hcju/proofwikiqa huggingface dataset:
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \operatorname{arcsec} \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = \begin{cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}       | c = Derivative of $\operatorname{arcsec} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = x^2       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {x^3} 3       | c = Primitive of Power }} {{end-eqn}} First let $\operatorname{arcsec} \dfrac x a$ be in the interval $\left({0 \,.\,.\,\dfrac \pi 2}\right)$. Then: {{begin-eqn}} {{eqn | l = \int x^2 \operatorname{arcsec} \frac x a \ \mathrm d x       | r = \frac {x^3} 3 \operatorname{arcsec} \frac x a - \int \frac {x^3} 3 \left({\frac a {x \sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^3} 3 \operatorname{arcsec} \frac x a - \frac a 3 \int \frac {x^2 \ \mathrm d x} {\sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {x^3} 3 \operatorname{arcsec} \frac x a - \frac a 3 \left({\frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \ln \left({x + \sqrt {x^2 - a^2} }\right)}\right) + C       | c = Primitive of $\dfrac {x^2} {\sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {x^3} 3 \operatorname{arcsec} \frac x a - \frac {a x \sqrt{x^2 - a^2} } 6 - \frac {a^3} 6 \ln \left({x + \sqrt {x^2 - a^2} }\right) + C       | c = simplifying }} {{end-eqn}} Similarly, let $\operatorname{arcsec} \dfrac x a$ be in the interval $\left({\dfrac \pi 2 \,.\,.\, \pi}\right)$. Then: {{begin-eqn}} {{eqn | l = \int x^2 \operatorname{arcsec} \frac x a \ \mathrm d x       | r = \frac {x^3} 3 \operatorname{arcsec} \frac x a - \int \frac {x^3} 3 \left({\frac {-a} {x \sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^3} 3 \operatorname{arcsec} \frac x a + \frac a 3 \int \frac {x^2 \ \mathrm d x} {\sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {x^3} 3 \operatorname{arcsec} \frac x a + \frac a 3 \left({\frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \ln \left({x + \sqrt {x^2 - a^2} }\right)}\right) + C       | c = Primitive of $\dfrac {x^2} {\sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {x^3} 3 \operatorname{arcsec} \frac x a + \frac {a x \sqrt{x^2 - a^2} } 6 + \frac {a^3} 6 \ln \left({x + \sqrt {x^2 - a^2} }\right) + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \operatorname{arcsec} \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = \begin{cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}       | c = Derivative of $\operatorname{arcsec} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = \frac 1 {x^2}       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {-1} x       | c = Primitive of Power }} {{end-eqn}} First let $\operatorname{arcsec} \dfrac x a$ be in the interval $\left({0 \,.\,.\,\dfrac \pi 2}\right)$. Then: {{begin-eqn}} {{eqn | l = \int \frac {\operatorname{arcsec} \frac x a} {x^2}       | r = \frac {-1} x \operatorname{arcsec} \frac x a - \int \frac {-1} x \left({\frac a {x \sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {-\operatorname{arcsec} \frac x a} x + a \int \frac {\mathrm d x} {x \sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {-\operatorname{arcsec} \frac x a} x + a \left({\frac {\sqrt {x^2 - a^2} } {a^2 x} }\right) + C       | c = Primitive of $\dfrac 1 {x^2 \sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {-\operatorname{arcsec} \frac x a} x + \frac {\sqrt{x^2 - a^2} } {a x} + C       | c = simplifying }} {{end-eqn}} Similarly, let $\operatorname{arcsec} \dfrac x a$ be in the interval $\left({\dfrac \pi 2 \,.\,.\, \pi}\right)$. Then: {{begin-eqn}} {{eqn | l = \int \frac {\operatorname{arcsec} \frac x a} {x^2}       | r = \frac {-1} x \operatorname{arcsec} \frac x a - \int \frac {-1} x \left({\frac {-a} {x \sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {-\operatorname{arcsec} \frac x a} x - a \int \frac {\mathrm d x} {x \sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {-\operatorname{arcsec} \frac x a} x - a \left({\frac {\sqrt {x^2 - a^2} } {a^2 x} }\right) + C       | c = Primitive of $\dfrac 1 {x^2 \sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {-\operatorname{arcsec} \frac x a} x - \frac {\sqrt{x^2 - a^2} } {a x} + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \arccsc \frac x a       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}       | c = Derivative of $\arccsc \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \frac 1 {x^2}       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {-1} x       | c = Primitive of Power }} {{end-eqn}} First let $\arccsc \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$. Then: {{begin-eqn}} {{eqn | l = \int \frac {\arccsc \frac x a} {x^2}       | r = \frac {-1} x \arccsc \frac x a - \int \frac {-1} x \paren {\frac {-a} {x \sqrt {x^2 - a^2} } } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {-\arccsc \frac x a} x - a \int \frac {\d x} {x \sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {-\arccsc \frac x a} x - a \paren {\frac {\sqrt {x^2 - a^2} } {a^2 x} } + C       | c = Primitive of $\dfrac 1 {x^2 \sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {-\arccsc \frac x a} x - \frac {\sqrt{x^2 - a^2} } {a x} + C       | c = simplifying }} {{end-eqn}} Similarly, let $\arccsc \dfrac x a$ be in the interval $\openint {-\dfrac \pi 2} 0$. Then: {{begin-eqn}} {{eqn | l = \int \frac {\arccsc \frac x a} {x^2}       | r = \frac {-1} x \arccsc \frac x a - \int \frac {-1} x \paren {\frac a {x \sqrt {x^2 - a^2} } } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {-\arccsc \frac x a} x + a \int \frac {\d x} {x \sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac {-\arccsc \frac x a} x + a \paren {\frac {\sqrt {x^2 - a^2} } {a^2 x} } + C       | c = Primitive of $\dfrac 1 {x^2 \sqrt {x^2 - a^2} }$ }} {{eqn | r = \frac {-\arccsc \frac x a} x + \frac {\sqrt{x^2 - a^2} } {a x} + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \arcsin \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\d u} {\d x}       | r = \frac 1 {\sqrt {a^2 - x^2} }       | c = Derivative of $\arcsin \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = x^m       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {x^{m + 1} } {m + 1}       | c = Primitive of Power }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x^m \arcsin \frac x a \rd x       | r = \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \int \frac {x^{m + 1} } {m + 1} \left({\frac 1 {\sqrt {a^2 - x^2} } }\right) \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^{m + 1} } {m + 1} \arcsin \frac x a - \frac 1 {m + 1} \int \frac {x^{m + 1} \rd x} {\sqrt{a^2 - x^2} } + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$ let: {{begin-eqn}} {{eqn | l = u       | r = \operatorname{arcsec} \frac x a       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = \begin{cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}       | c = Derivative of $\operatorname{arcsec} \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm d x}       | r = x^m       | c =  }} {{eqn | ll= \implies       | l = v       | r = \frac {x^{m + 1} } {m + 1}       | c = Primitive of Power }} {{end-eqn}} First let $\operatorname{arcsec} \dfrac x a$ be in the interval $\left({0 \,.\,.\,\dfrac \pi 2}\right)$. Then: {{begin-eqn}} {{eqn | l = \int x^m \operatorname{arcsec} \frac x a \ \mathrm d x       | r = \frac {x^{m + 1} } {m + 1} \operatorname{arcsec} \frac x a - \int \frac {x^{m + 1} } {m + 1} \left({\frac a {x \sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^{m + 1} } {m + 1} \operatorname{arcsec} \frac x a - \frac a {m + 1} \int \frac {x^m \ \mathrm d x} {\sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} Similarly, let $\operatorname{arcsec} \dfrac x a$ be in the interval $\left({\dfrac \pi 2 \,.\,.\, \pi}\right)$. Then: {{begin-eqn}} {{eqn | l = \int x^m \operatorname{arcsec} \frac x a \ \mathrm d x       | r = \frac {x^{m + 1} } {m + 1} \operatorname{arcsec} \frac x a - \int \frac {x^{m + 1} } {m + 1} \left({\frac {-a} {x \sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^{m + 1} } {m + 1} \operatorname{arcsec} \frac x a + \frac a {m + 1} \int \frac {x^m \ \mathrm d x} {\sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\rd v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = \arccsc \frac x a       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = \begin {cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\ \dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\ \end{cases}       | c = Derivative of $\arccsc \dfrac x a$ }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = x^m       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {x^{m + 1} } {m + 1}       | c = Primitive of Power }} {{end-eqn}} First let $\arccsc \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$. Then: {{begin-eqn}} {{eqn | l = \int x^m \arccsc \frac x a \rd x       | r = \frac {x^{m + 1} } {m + 1} \arccsc \frac x a - \int \frac {x^{m + 1} } {m + 1} \paren {\frac {-a} {x \sqrt {x^2 - a^2} } } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^{m + 1} } {m + 1} \arccsc \frac x a + \frac a {m + 1} \int \frac {x^m \rd x} {\sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} Similarly, let $\arccsc \dfrac x a$ be in the interval $\openint {-\dfrac \pi 2}  0$. Then: {{begin-eqn}} {{eqn | l = \int x^m \arccsc \frac x a \rd x       | r = \frac {x^{m + 1} } {m + 1} \arccsc \frac x a - \int \frac {x^{m + 1} } {m + 1} \paren {\frac a {x \sqrt {x^2 - a^2} } } \rd x + C       | c = Integration by Parts }} {{eqn | r = \frac {x^{m + 1} } {m + 1} \arccsc \frac x a - \frac a {m + 1} \int \frac {x^m \rd x} {\sqrt {x^2 - a^2} } + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int e^{a x} \sin b x \rd x       | r = \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x       | c = Primitive of $e^{a x} \sin b x$: Lemma }} {{eqn | r = \frac {e^{a x} \sin b x} a - \frac b a \paren {\frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x}       | c = Primitive of $e^{a x} \cos b x$: Lemma }} {{eqn | r = \frac {e^{a x} a \sin b x - e^{a x} b \cos b x} {a^2} - \frac {b^2} {a^2} \int e^{a x} \sin b x \rd x       | c = simplifying }} {{eqn | ll= \leadsto       | l = \paren {1 + \frac {b^2} {a^2} } \int e^{a x} \sin b x \rd x       | r = \frac {e^{a x} \left({a \sin b x - b \cos b x}\right)} {a^2}        | c = simplifying }} {{eqn | ll= \leadsto       | l = \frac {a^2 + b^2} {a^2} \int e^{a x} \sin b x \rd x       | r = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2}        | c = common denominator }} {{eqn | ll= \leadsto       | l = \int e^{a x} \sin b x \rd x       | r = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2}        | c = multiplying by $\dfrac {a^2} {a^2 + b^2}$ }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \cos b x + i \sin b x       | r = e^{i b x}       | c = Euler's Formula }} {{eqn | ll= \leadsto       | l = e^{a x} \cos b x + i e^{a x} \sin b x       | r = e^{a x} e^{i b x}       | c = multiplying both sides by $e^{a x}$ }} {{eqn | r = e^{\paren {a + i b} x}       | c = Exponent Combination Laws }} {{eqn | ll= \leadsto       | l = \int e^{a x} \cos b x \, \d x + i \int e^{a x} \sin b x \d x       | r = \int e^{\paren {a + i b} x} \d x       | c = Linear Combination of Complex Integrals }} {{eqn | r = \frac 1 {a + i b} e^{\paren {a + i b} x} + C       | c = Primitive of $e^{a x}$ }} {{eqn | r = \frac {a - i b} {a^2 + b^2} e^{\paren {a + i b} x} + C       | c = multiplying top and bottom by $a - i b$ }} {{eqn | r = \frac {a - i b} {a^2 + b^2} e^{a x} e^{i b x} + C       | c = Exponent Combination Laws }} {{eqn | r = \frac {a - i b} {a^2 + b^2} e^{a x} \paren {\cos b x + i \sin b x} + C       | c = Euler's Formula }} {{eqn | r = \frac a {a^2 + b^2} e^{a x} \cos b x - \frac {i b} {a^2 + b^2} e^{a x} \cos b x }} {{eqn | o =       | ro= +       | r =  \frac {i a} {a^2 + b^2} e^{a x} \sin b x + \frac b {a^2 + b^2} e^{a x} \sin b x + C }} {{end-eqn}} The result follows from equating imaginary parts. {{qed}}
{{begin-eqn}} {{eqn | l = \int e^{a x} \sin b x \rd x       | r = \int e^{a x} \paren {\frac {e^{i b x} - e^{-i b x} } {2 i} } \rd x       | c = Sine Exponential Formulation }} {{eqn | r = \frac 1 {2 i} \int e^{a x} \paren {e^{i b x} - e^{-i b x} } \rd x       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 {2 i} \int \paren {e^{a x} e^{i b x} - e^{a x} e^{-i b x} } \rd x       | c =  }} {{eqn | r = \frac 1 {2 i} \int \paren {e^{a x + i b x} - e^{a x - i b x} } \rd x       | c = Exponent Combination Laws: Product of Powers }} {{eqn | r = \frac 1 {2 i} \int e^{a x + i b x} \rd x - \frac 1 {2 i} \int e^{a x - i b x} \rd x       | c = Linear Combination of Integrals }} {{eqn | r = \frac 1 {2 i} \int e^{\paren {a + i b} x} \rd x - \frac 1 {2 i} \int e^{\paren {a - i b} x} \rd x       | c =  }} {{eqn | r = \frac 1 {2 i} \frac {e^{\paren {a + i b} x} } {a + i b} - \frac 1 {2 i} \frac {e^{\paren {a - i b} x} } {a - i b} + C       | c = Primitive of $e^{a x}$ }} {{eqn | r = \frac 1 {2 i} \frac {e^{a x + i b x} } {a + i b} - \frac 1 {2 i} \frac {e^{a x - i b x} } {a - i b} + C       | c =  }} {{eqn | r = \frac 1 {2 i} \frac {e^{a x} e^{i b x} } {a + i b} - \frac 1 {2 i} \frac {e^{a x} e^{-i b x} } {a - i b} + C       | c = Exponent Combination Laws: Product of Powers }} {{eqn | r = \frac 1 {2 i} \frac {e^{a x} e^{i b x} \paren {a - i b} } {\paren {a + i b} \paren {a - i b} } - \frac 1 {2 i} \frac {e^{a x} e^{-i b x} \paren {a + i b} } {\paren {a - i b} \paren {a + i b} } + C       | c =  }} {{eqn | r = \frac {e^{a x} e^{i b x} \paren {a - i b} - e^{a x} e^{-i b x} \paren {a + i b} } {2 i \paren {a + i b} \paren {a - i b} } + C       | c =  }} {{eqn | r = \frac {e^{a x} e^{i b x} \paren {a - i b} - e^{a x} e^{-i b x} \paren {a + i b} } {2 i \paren {a^2 + b^2} } + C       | c = Product of Complex Number with Conjugate }} {{eqn | r = \frac {a e^{a x} e^{i b x} - i b e^{a x} e^{i b x} - a e^{a x} e^{-i b x} - i b e^{a x} e^{-i b x} } {2 i \paren {a^2 + b^2} } + C       | c =  }} {{eqn | r = \frac {e^{a x} } {\paren {a^2 + b^2} } \paren {\frac {a e^{i b x} - i b e^{i b x} - a e^{-i b x} - i b e^{-i b x} } {2 i} } + C       | c =  }} {{eqn | r = \frac {e^{a x} } {\paren {a^2 + b^2} } \paren {a \frac {e^{i b x} - e^{-i b x} } {2 i} - b \frac {e^{i b x} + e^{-i b x} } 2 } + C       | c =  }} {{eqn | r = \frac {e^{a x} } {\paren {a^2 + b^2} } \paren {a \frac {e^{i b x} - e^{-i b x} } {2 i} - b \cos b x } + C       | c = Cosine Exponential Formulation }} {{eqn | r = \frac {e^{a x} } {\paren {a^2 + b^2} } \paren {a \sin b x - b \cos b x } + C       | c = Sine Exponential Formulation }} {{eqn | r = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C       | c =  }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn	| l = \int e^{a x} e^{i b x} \rd x 	| r = i \int e^{a x} \sin b x \rd x + \int e^{a x} \cos b x \rd x 	| c = Euler's Formula }} {{eqn	| ll= \leadsto 	| l = \int e^{a x} \cos b x \rd x 	| r = \map \Re {\int e^{\paren {a + i b} x} \rd x}  }} {{eqn	| r = \map \Re {\frac {e^{\paren {a + i b} x} } {a + i b} } + C 	| c = Primitive of Exponential of a x }} {{eqn	| r = \map \Re {\frac {\paren {a - i b}  e^{\paren {a + i b} x} } {a^2 + b^2} } + C 	| c = multiplying through by $\dfrac {a - i b} {a - i b}$ }} {{eqn	| r = \map \Re {\frac {i a e^{a x} \sin b x + a e^{a x} \cos b x - i b \paren {i e^{a x} \sin b x + e^{a x} \cos b x} } {a^2 + b^2} } + C 	| c = Euler's Formula }} {{eqn	| r = \map \Re {\frac {i \paren {a e^{a x} \sin b x - b e^{a x} \cos b x} + \paren {a e^{a x} \cos b x + b e^{a x} \sin b x} } { a^2 + b^2} } + C }} {{eqn	| r = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2} + C 	| c = isolating real part }} {{end-eqn}} {{qed}}
{{begin-eqn}} {{eqn | l = \int e^{a x} \cos b x \rd x       | r = \frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x       | c = Primitive of $e^{a x} \cos b x$: Lemma }} {{eqn | r = \frac {e^{a x} \cos b x} a + \frac b a \paren {\frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x}       | c = Primitive of $e^{a x} \sin b x$: Lemma }} {{eqn | r = \frac {e^{a x} a \cos b x + e^{a x} b \sin b x} {a^2} - \frac {b^2} {a^2} \int e^{a x} \cos b x \rd x       | c = simplifying }} {{eqn | ll= \leadsto       | l = \paren {1 + \frac {b^2} {a^2} } \int e^{a x} \cos b x \rd x       | r = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2}        | c = simplifying }} {{eqn | ll= \leadsto       | l = \frac {a^2 + b^2} {a^2} \int e^{a x} \cos b x \rd x       | r = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2}        | c = common denominator }} {{eqn | ll= \leadsto       | l = \int e^{a x} \cos b x \rd x       | r = \frac {e^{a x} \paren {a \cos b x + b \sin b x} } {a^2 + b^2}        | c = multiplying by $\dfrac {a^2} {a^2 + b^2}$ }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = 1       | c = Derivative of Identity Function }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = e^{a x}       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {e^{a x} } a       | c = Primitive of $e^{a x}$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x e^{a x} \rd x       | r = x \paren {\frac {e^{a x} } a} - \int \frac {e^{a x} } a \rd x + C       | c = Integration by Parts }} {{eqn | r = x \paren {\frac {e^{a x} } a} - \frac 1 a \int e^{a x} \rd x + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = x \paren {\frac {e^{a x} } a} - \frac 1 a \paren {\frac {e^{a x} } a} + C       | c = Primitive of $e^{a x}$ }} {{eqn | r = \frac {e^{a x} } a \paren {x - \frac 1 a} + C       | c = simplifying }} {{end-eqn}} {{qed}}
With a view to expressing the primitive in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x^2       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = 2 x       | c = Derivative of Power }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = e^{a x}       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {e^{a x} } a       | c = Primitive of $e^{a x}$ }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | l = \int x e^{a x} \rd x       | r = x^2 \paren {\frac {e^{a x} } a} - \int 2 x \frac {e^{a x} } a \rd x + C       | c = Integration by Parts }} {{eqn | r = x^2 \paren {\frac {e^{a x} } a} - \frac 2 a \int x e^{a x} \rd x + C       | c = Primitive of Constant Multiple of Function }} {{eqn | r = x^2 \paren {\frac {e^{a x} } a} - \frac 2 a \paren {\frac {e^{a x} } a \paren {x - \frac 1 a} } + C       | c = Primitive of $x e^{a x}$ }} {{eqn | r = \frac {e^{a x} } a \paren {x^2 - \frac {2 x} a + \frac 2 {a^2} } + C       | c = simplifying }} {{end-eqn}} {{qed}}