Evaluate the integral. (Hint: you may need to use a reduction formula.)

Question

Quizplus · Accepted Answer

The integral requires using a reduction formula for integrals of the form $\int e^{ax}\sin^nx\ dx$. By applying this reduction formula twice, the integral reduces to $\frac{1}{25}(11e^{-x}\sin^3x+3e^{-x}\sin x-\cos x)+C$. Plugging in $x=0$ yields $\frac{1}{25}(11\cdot 0^3+3\cdot 0-1)+C=-\frac{1}{25}+C$. Finally, adding the constant of integration gives the answer $\frac{11}{25}e^{-x}\sin^3x+\frac{3}{25}e^{-x}\sin x-\frac{1}{25}+	ilde{C}$, where $	ilde{C}$ is the constant that includes the original constant of integration.

Evaluate the Integral