zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of 4sin^3(2x)

解答

\int 4 sin^{3} (2 x) d x

解答

32 (- \frac{1}{4} cos^{4} (x) + \frac{1}{6} cos^{6} (x)) + C

求解步骤

\int 4 sin^{3} (2 x) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int sin^{3} (2 x) d x

使用三角恒等式改写

= 4 \cdot \int 8 cos^{3} (x) sin^{3} (x) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot 8 \cdot \int cos^{3} (x) sin^{3} (x) d x

使用三角恒等式改写

= 4 \cdot 8 \cdot \int (1 - cos^{2} (x)) sin (x) cos^{3} (x) d x

使用换元积分法

= 4 \cdot 8 \cdot \int - u^{3} (1 - u^{2}) d u

乘开

- u^{3} (1 - u^{2}) : - u^{3} + u^{5}

= 4 \cdot 8 \cdot \int - u^{3} + u^{5} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 \cdot 8 (- \int u^{3} d u + \int u^{5} d u)

\int u^{3} d u = \frac{u ^{4}}{4}

\int u^{5} d u = \frac{u ^{6}}{6}

= 4 \cdot 8 (- \frac{u ^{4}}{4} + \frac{u ^{6}}{6})

u = cos (x)

代回

= 4 \cdot 8 (- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{6})

化简

4 \cdot 8 (- \frac{cos ^{4} ( x )}{4} + \frac{cos ^{6} ( x )}{6}) : 32 (- \frac{1}{4} cos^{4} (x) + \frac{1}{6} cos^{6} (x))

= 32 (- \frac{1}{4} cos^{4} (x) + \frac{1}{6} cos^{6} (x))

解答补常数

= 32 (- \frac{1}{4} cos^{4} (x) + \frac{1}{6} cos^{6} (x)) + C

作图

流行的例子

integral of 13-x

\int 13 - x d x

limit as x approaches-1 of 4/(x^2+2x+3)

x \to - 1 lim (\frac{4}{x ^{2} + 2 x + 3})

integral of (pisin(x)+sqrt(2)cos(x))

\int (π sin (x) + 2 cos (x)) d x

integral of sqrt(3-x)*x^2

\int 3 - x \cdot x^{2} d x

integral of (3xsqrt(4x^2-6))/5

\int \frac{3 x 4 x ^{2} - 6}{5} d x