題目

3. (10 points) Compute the following integrals

(a) (5 points) $\int_{0}^{\pi/2}\dfrac{\sin 2x}{2+\cos x}\,dx$ .

(b) (5 points) $\int_{1}^{2}\dfrac{(\ln x)^2}{x^3}\,dx$ .

解答

(a) 令 $u=\cos x$ ，則 $\mathrm{d}u=-\sin x\,\mathrm{d}x$ ：

\int_0^{\pi/2}\frac{\sin2x}{2+\cos x}\,\mathrm{d}x =\int_0^1\frac{2u}{2+u}\,\mathrm{d}u =\boxed{2-4\ln\frac32}.

(b) 對 $\int(\ln x)^2x^{-3}\,\mathrm{d}x$ 兩次分部積分，可得

\int\frac{(\ln x)^2}{x^3}\,\mathrm{d}x =-\frac{(\ln x)^2}{2x^2}-\frac{\ln x}{2x^2}-\frac1{4x^2}+C.

故所求值為

\boxed{\frac3{16}-\frac{(\ln2)^2+\ln2}{8}}.