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受欢迎的微积分 >

integral of cos^2(6x)sin^3(6x)

解答

\int cos^{2} (6 x) sin^{3} (6 x) d x

解答

\frac{1}{6} (- \frac{cos ^{3} ( 6 x )}{3} + \frac{cos ^{5} ( 6 x )}{5}) + C

求解步骤

\int cos^{2} (6 x) sin^{3} (6 x) d x

使用换元积分法

= \int cos^{2} (u) sin^{3} (u) \frac{1}{6} d u

提出常数:

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

= \frac{1}{6} \cdot \int cos^{2} (u) sin^{3} (u) d u

使用三角恒等式改写

= \frac{1}{6} \cdot \int (1 - cos^{2} (u)) sin (u) cos^{2} (u) d u

使用换元积分法

= \frac{1}{6} \cdot \int - v^{2} (1 - v^{2}) d v

乘开

- v^{2} (1 - v^{2}) : - v^{2} + v^{4}

= \frac{1}{6} \cdot \int - v^{2} + v^{4} d v

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

= \frac{1}{6} (- \int v^{2} d v + \int v^{4} d v)

\int v^{2} d v = \frac{v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= \frac{1}{6} (- \frac{v ^{3}}{3} + \frac{v ^{5}}{5})

代回

= \frac{1}{6} (- \frac{cos ^{3} ( 6 x )}{3} + \frac{cos ^{5} ( 6 x )}{5})

解答补常数

= \frac{1}{6} (- \frac{cos ^{3} ( 6 x )}{3} + \frac{cos ^{5} ( 6 x )}{5}) + C

作图

流行的例子

integral of (10)/(sqrt(5-x))

\int \frac{10}{5 - x} d x

麦克劳林 1/((1-x)^2)

ma c l a u r in \frac{1}{( 1 - x ) ^{2}}

(\partial)/(\partial x)(x^2+xy+5y)

\frac{\partial}{\partial x} (x^{2} + x y + 5 y)

limit as x approaches 2 of 6x^2

x \to 2 lim (6 x^{2})

derivative of (x^2+sqrt(x)/(2x+5))

\frac{d}{d x} (\frac{x ^{2} + x}{2 x + 5})