16.2 The Substitution Rules of Integration 换元积分法 - 独中课程 | 线上补习

高二数学 | 高级数学

About Lesson

16.2 The Substitution Rules of Integration 换元积分法

Evaluate the following indefinite integral
求下列不定积分

1) $\int_{0}^{2}\frac{xdx}{\sqrt{4x^2+9}}$

Answer :

$\frac{1}{2}$

2) $\int_{0}^{2}{x^2(1+2x^3)^3dx}$

Answer :

3480

3) $\int_{0}^{2}\frac{xdx}{\sqrt{2x^2+1}}$

Answer :

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

Answer :

$4\frac{2}{3}$

5) $\int_{0}^{4}{x\sqrt{16-3x}dx}$

Answer :

$\frac{3008}{135}$

*** QuickLaTeX cannot compile formula:
\int_{\sfrac{4}{\pi}}^{\sfrac{2}{\pi}}{\frac{1}{x^2}cos{\frac{2}{x}}dx}

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

Answer :

$\frac{1}{2}$

7) $\int_{1}^{2}{(2x+1)\sqrt{x-1}dx}$

Answer :

2.8

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

(a) $\int_{0}^{1}{f(2x)dx}$

(b) If f (-x) = f (x), find the value of $\int_{-2}^{0}{f(x)dx}$ .

Answer :

(a) 2.5

(b) 5

9) If f is continuous and $\int_{0}^{4}{f(x)dx}$ =10,find

(a) $\int_{0}^{2}{f(2x)dx}$

(b) $\int_{0}^{2}{xf(x^2)dx}$

Answer :

(a) 5

(b) 5