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 = \sin^n b x       | c =  }} {{eqn | ll= \implies       | l = \frac {\mathrm d u} {\mathrm d x}       | r = n b \sin^{n - 1} b x \cos b x       | c = Derivative of $\sin a x$, 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= \implies       | 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} \sin^n b x \ \mathrm d x       | r = \sin^n b x \left({\frac {e^{a x} } a}\right) - \int \left({\frac {e^{a x} } a}\right) n b \sin^{n - 1} b x \cos b x \ \mathrm d x + C       | c = Integration by Parts }} {{eqn | n = 1       | r = \frac {e^{a x} \sin^n b x} a - \frac {n b} a \int e^{a x} \sin^{n - 1} b x \cos b x \ \mathrm d x + C       | c = Primitive of Constant Multiple of Function }} {{end-eqn}} From Primitive of $e^{a x} \sin^{n - 1} b x \cos b x$: Lemma 1: :$\displaystyle \int e^{a x} \sin^{n - 1} b x \cos b x \ \mathrm d x = \frac {e^{a x} \sin^{n - 1} b x \left({a \cos b x + b \sin b x}\right)} {a^2 + n b^2} + \frac {\left({n - 1}\right) a b} {a^2 + n b^2} \left({\int e^{a x} \sin^n b x \ \mathrm d x - \int e^{a x} \sin^{n - 2} b x \ \mathrm d x}\right) + C$ Hence: {{begin-eqn}} {{eqn | r = \int e^{a x} \sin^n b x \ \mathrm d x       | o =        | c =  }} {{eqn | r = \frac {e^{a x} \sin^n b x} a - \frac {n b} a \int e^{a x} \sin^{n - 1} b x \cos b x \ \mathrm d x + C       | c = from $(1)$ }} {{eqn | r = \frac {e^{a x} \sin^n b x} a - \frac {n b} {a \left({a^2 + n b^2}\right)} e^{a x} \sin^{n - 1} b x \left({a \cos b x + b \sin b x}\right)       | c = Primitive of $e^{a x} \sin^n b x$: Lemma 1 }} {{eqn | o =        | ro= -       | r = \frac {n \left({n - 1}\right) b^2} {a^2 + n b^2} \int e^{a x} \sin^n b x \ \mathrm d x + \frac {n \left({n - 1}\right) b^2} {a^2 + n b^2} \int e^{a x} \sin^{n - 2} b x \ \mathrm d x + C       | c =  }} {{eqn | ll= \implies       | r = \left({1 + \frac {n \left({n - 1}\right) b^2} {a^2 + n b^2} }\right) \int e^{a x} \sin^n b x \ \mathrm d x       | o =        | c =  }} {{eqn | r = \frac {e^{a x} \sin^n b x} a - \frac {n b} {a \left({a^2 + n b^2}\right)} e^{a x} \sin^{n - 1} b x \left({a \cos b x + b \sin b x}\right)       | c =  }} {{eqn | o =        | ro= +       | r = \frac {n \left({n - 1}\right) b^2} {a^2 + n b^2} \int e^{a x} \sin^{n - 2} b x \ \mathrm d x + C       | c =  }} {{eqn | ll= \implies       | r = \frac {a^2 + n b^2 + n^2 b^2 - n b^2} {a^2 + n b^2} \int e^{a x} \sin^n b x \ \mathrm d x       | o =        | c =  }} {{eqn | r = \frac {e^{a x} \sin^n b x} a - \frac {n b} {a \left({a^2 + n b^2}\right)} e^{a x} \sin^{n - 1} b x \left({a \cos b x + b \sin bx}\right)       | c =  }} {{eqn | o =        | ro= +       | r = \frac {n \left({n - 1}\right) b^2} {a^2 + n b^2} \int e^{a x} \sin^{n - 2} b x \ \mathrm d x + C       | c =  }} {{eqn | ll= \implies       | r = \left({a^2 + n^2 b^2}\right) \int e^{a x} \sin^n b x \ \mathrm d x       | o =        | c =  }} {{eqn | r = \frac {a^2 + n b^2} a e^{a x} \sin^n b x - \frac {n b} a e^{a x} \sin^{n - 1} b x \left({a \cos b x + b \sin b x}\right)       | c =  }} {{eqn | o =        | ro= +       | r = \left({n \left({n - 1}\right) b^2}\right) \int e^{a x} \sin^{n - 2} b x \ \mathrm d x + C       | c =  }} {{eqn | ll= \implies       | r = \left({a^2 + n^2 b^2}\right) \int e^{a x} \sin^n b x \ \mathrm d x       | o =        | c =  }} {{eqn | r = e^{a x} \sin^{n - 1} b x \left({a \sin b x - n b \cos b x}\right)       | c = Primitive of $e^{a x} \sin^n b x$: Lemma 2 }} {{eqn | o =        | ro= +       | r = \left({n \left({n - 1}\right) b^2}\right) \int e^{a x} \sin^{n - 2} b x \ \mathrm d x + C       | c =  }} {{eqn | ll= \implies       | r = \int e^{a x} \sin^n b x \ \mathrm d x       | o =        | c =  }} {{eqn | r = \frac {e^{a x} \sin^{n - 1} b x} {a^2 + n^2 b^2} \left({a \sin b x - n b \cos b x}\right) + \frac {n \left({n - 1}\right) b^2} {a^2 + n^2 b^2} \int e^{a x} \sin^{n - 2} b x \ \mathrm d x + C       | c =  }} {{end-eqn}} {{qed}} {{Proofread}}