第9章 9.1

(1)

$\int_0^2 (x^3-3x^2)dx = \left[\frac{1}{4}x^4-x^3\right]_0^2 = \frac{1}{4}\cdot16-8=4-8=-4$

(2)

$\int_2^4 (x^3-4x+5)dx = \left[\frac{1}{4}x^4-2x^2+5x\right]_2^4 = \frac{1}{4}\cdot256-2\cdot16+5\cdot4-\frac{1}{4}\cdot16+2\cdot4-5\cdot2$

$=64-32+20-4+8-10=46$

(3)

$\int_0^9 \sqrt{x}\,dx = \left[\frac{2}{3}\sqrt{x^3}\right]_0^9 = \frac{2}{3}(27-0)=18$

(4)

$\int_1^2 \frac{1}{\sqrt{x}}\,dx = \left[2\sqrt{x}\right]_1^2 = 2(\sqrt{2}-1)$

(5)

$\int_0^{\frac{\pi}{4}} \cos x\,dx = \left[\sin x\right]_0^{\frac{\pi}{4}} = \sin\frac{\pi}{4}-0=\frac{1}{\sqrt{2}}$

(6)

$\int_0^1 (1-\sqrt{x})^2\,dx = \int_0^1 (1-2\sqrt{x}+x)dx = \left[x-\frac{4}{3}\sqrt{x^3}+\frac{1}{2}x^2\right]_0^1 = 1-\frac{4}{3}+\frac{1}{2}=\frac{1}{6}$

(7)

$\int_{-2}^0 \frac{1}{x+3}\,dx = \left[\log(x+3)\right]_{-2}^0 = \log 3-\log 1=\log 3$

(8)

$\int_0^1 \frac{e^x+e^{-x}}{2}dx=\left[\frac{e^x-e^{-x}}{2}\right]_0^1=\frac{e-e^{-1}-1+1}{2}=\frac{1}{2}\left(e-\frac{1}{e}\right)$

(9)

$\int_1^3\left(e^x+\frac{1}{x}\right)dx=\left[e^x+\log x\right]_1^3=e^3-\log 3-e-0=e^3-e-\log 3$

(10)

$\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{1}{\cos^2 x}dx=\left[\tan x\right]_{\frac{\pi}{6}}^{\frac{\pi}{4}}=\tan\frac{\pi}{4}-\tan\frac{\pi}{6}=1-\frac{1}{\sqrt{3}}$