16.1 Fundamental Definite Integral 基本定积分 - 独中课程 | 线上补习

高二数学 | 高级数学

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16.1 Fundamental Definite Integral 基本定积分

Evaluate the following definite integral
求下列定积分

1) $\int_{1}^{4}{\frac{5x^2+3x+1}{\sqrt x}dx}$

Answer :

2) $\int_{1}^{8}\sqrt[3]{x}(x-1)dx$

Answer :

$43\frac{5}{28}$

3) $\int_{1}^{2}{(2x-3)^{20}dx}$

Answer :

$\frac{1}{21}$

4) $\int_{0}^{1}{\frac{2x+1}{(4x^2+4x+1)^2}dx}$

Answer :

$\frac{2}{9}$

5) $\int_{0}^{4}{|x^2-4|dx}$

Answer :

6) $\int_{0}^{3}{sin{(}\frac{2\pi x}{3}+\frac{\pi}{4})dx}$

Answer :

*** QuickLaTeX cannot compile formula:
\int_{\sfrac{\pi}{4}}^{\sfrac{\pi}{3}}{(1-{tan}^2{x})dx}

*** Error message:
Undefined control sequence \sfrac.
leading text: $\int_{\sfrac

Answer :

$1 + \frac{\pi}{6}-\sqrt3$

*** QuickLaTeX cannot compile formula:
\int_{0}^{\sfrac{\pi}{3}}\frac{1-{tan}^2{x}}{{sec}^2{x}}dx

*** Error message:
Undefined control sequence \sfrac.
leading text: $\int_{0}^{\sfrac

Answer :

$\frac{\sqrt3}{4}$

9) $\int_{0}^{\pi}{{sin}^2{2}xdx}$

Answer :

$\frac{\pi}{2}$

10) Given that (已知) $\int_{0}^{2}{f(x)dx}$ = $\int_{2}^{3}{f(x)dx}$ = 5. Calculate (计算)

(a) $\int_{0}^{3}{f(x)dx}$

(b) $\int_{0}^{2}{f(x)dx+\int_{3}^{2}{f(x)dx}}$

Answer :

(a) 10

(b) 0