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受欢迎的微积分 >

integral of (-sin(3x))/(cos^3(3x))

解答

\int \frac{- sin ( 3 x )}{cos ^{3} ( 3 x )} d x

解答

- \frac{1}{6} tan^{2} (3 x) + C

求解步骤

\int \frac{- sin ( 3 x )}{cos ^{3} ( 3 x )} d x

提出常数:

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

= - \int \frac{sin ( 3 x )}{cos ^{3} ( 3 x )} d x

使用三角恒等式改写

= - \int \frac{tan ( 3 x )}{cos ^{2} ( 3 x )} d x

使用基本三角恒等式:

\frac{1}{cos ( x )} = sec (x)

= - \int sec^{2} (3 x) tan (3 x) d x

使用换元积分法

= - \int \frac{u}{3} d u

提出常数:

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

= - \frac{1}{3} \cdot \int u d u

使用幂法则

= - \frac{1}{3} \cdot \frac{u ^{2}}{2}

u = tan (3 x)

代回

= - \frac{1}{3} \cdot \frac{tan ^{2} ( 3 x )}{2}

化简

- \frac{1}{3} \cdot \frac{tan ^{2} ( 3 x )}{2} : - \frac{1}{6} tan^{2} (3 x)

= - \frac{1}{6} tan^{2} (3 x)

解答补常数

= - \frac{1}{6} tan^{2} (3 x) + C

作图

流行的例子

derivative of sqrt(-2x^2+1)

\frac{d}{d x} (- 2 x^{2} + 1)

derivative of 4csc(x)

\frac{d}{d x} (4 csc (x))

derivative of (x^{3/2}+2/x)

\frac{d}{d x} (\frac{x ^{\frac{3}{2}} + 2}{x})

integral of 1/(cos^2(x)tan(x))

\int \frac{1}{cos ^{2} ( x ) tan ( x )} d x

integral from 0 to sqrt(pi of)5xcos(x^2)

\int_{0}^{π} 5 x cos (x^{2}) d x