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 = \cos^n b x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = -n b \cos^{n - 1} b x \sin b x       | c = Derivative of $\cos a x$, Derivative of Power, Chain Rule for Derivatives }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\mathrm d v} {\mathrm 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 e^{a x} \cos^n b x \rd x       | r = \cos^n b x \paren {\frac {e^{a x} } a} - \int \paren {\frac {e^{a x} } a} \paren {-n b \cos^{n - 1} b x \sin b x} \rd x + C       | c = Integration by Parts }} {{eqn | n = 1       | r = \frac {e^{a x} \cos^n b x} a + \frac {n b} a \int e^{a x} \cos^{n - 1} b x \sin b x \rd x + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} From Primitive of $e^{a x} \cos^{n - 1} b x \sin b x$: :$\displaystyle \int e^{a x} \cos^{n - 1} b x \sin b x \rd x = \frac {e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x} } {a^2 + n b^2} - \frac {\paren {n - 1} a b} {a^2 + n b^2} \paren {\int e^{a x} \cos^n b x \rd x - \int e^{a x} \cos^{n - 2} b x \rd x} + C$ Hence: {{begin-eqn}} {{eqn | r = \int e^{a x} \cos^n b x \rd x       | o =        | c =  }} {{eqn | r = \frac {e^{a x} \cos^n b x} a + \frac {n b} a \int e^{a x} \cos^{n - 1} b x \sin b x \rd x + C       | c = from $(1)$ }} {{eqn | r = \frac {e^{a x} \cos^n b x} a + \frac {n b} {a \paren {a^2 + n b^2} } e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}       | c = Primitive of $e^{a x} \cos^n b x$: Lemma 1 }} {{eqn | o =        | ro= -       | r = \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^n b x \rd x + \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C       | c =  }} {{eqn | ll= \leadsto       | r = \paren {1 + \frac {n \paren {n - 1} b^2} {a^2 + n b^2} } \int e^{a x} \cos^n b x \rd x       | o =        | c =  }} {{eqn | r = \frac {e^{a x} \cos^n b x} a + \frac {n b} {a \paren {a^2 + n b^2} } e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}       | c =  }} {{eqn | o =        | ro= +       | r = \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C       | c =  }} {{eqn | ll= \leadsto       | r = \frac {a^2 + n b^2 + n^2 b^2 - n b^2} {a^2 + n b^2} \int e^{a x} \cos^n b x \rd x       | o =        | c =  }} {{eqn | r = \frac {e^{a x} \cos^n b x} a + \frac {n b} {a \paren {a^2 + n b^2} } e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}       | c =  }} {{eqn | o =        | ro= +       | r = \frac {n \paren {n - 1} b^2} {a^2 + n b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C       | c =  }} {{eqn | ll= \leadsto       | r = \paren {a^2 + n^2 b^2} \int e^{a x} \cos^n b x \rd x       | o =        | c =  }} {{eqn | r = \frac {a^2 + n b^2} a e^{a x} \cos^n b x + \frac {n b} a e^{a x} \cos^{n - 1} b x \paren {a \sin b x - b \cos b x}       | c =  }} {{eqn | o =        | ro= +       | r = \paren {n \paren {n - 1} b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C       | c =  }} {{eqn | ll= \leadsto       | r = \paren {a^2 + n^2 b^2} \int e^{a x} \cos^n b x \rd x       | o =        | c =  }} {{eqn | r = e^{a x} \cos^{n - 1} b x \paren {a \cos b x + n b \sin b x}       | c = Primitive of $e^{a x} \cos^n b x$: Lemma 2 }} {{eqn | o =        | ro= +       | r = \paren {n \paren {n - 1} b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C       | c =  }} {{eqn | ll= \leadsto       | r = \int e^{a x} \cos^n b x \rd x       | o =        | c =  }} {{eqn | r = \frac {e^{a x} \cos^{n - 1} b x} {a^2 + n^2 b^2} \paren {a \cos b x + n b \sin b x} + \frac {n \paren {n - 1} b^2} {a^2 + n^2 b^2} \int e^{a x} \cos^{n - 2} b x \rd x + C       | c =  }} {{end-eqn}} {{qed}} {{Proofread}}