Review question

# How could we integrate $e^{-x}\sin^n x$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8134

## Suggestion

1. Show that $\dfrac{d}{dx} (\cos x \sin^{n-1}x) = (n-1) \sin^{n-2}x - n \sin^{n}x$.

Would it help us to use $s$ for $\sin x$ and $c$ for $\cos x$?

How can we differentiate a product?

How could we convert any $\cos x$ terms into $\sin x$ terms?

1. Given that $I_n = \displaystyle\int_0^{\frac{1}{2}\pi} e^{-x} \sin^n x \, dx$, show that $I_n = -e^{-\frac{1}{2}\pi} + n \int_0^{\frac{1}{2}\pi} e^{-x} \cos x \sin^{n-1}x \, dx, \quad (n \ge 1).$

Do we know a ‘product rule’ for integration?

1. By using the results of (i) and (ii), or otherwise, show that $(n^2 + 1) I_n = -e^{-\frac{1}{2}\pi} + n(n-1) I_{n-2}, \quad (n \ge 2).$

Could we use our ‘product rule’ for integration a second time?

1. Show that $I_4 = \frac{1}{85} (24 - 41 e^{-\frac{1}{2}\pi})$.

How would the earlier parts help us here?