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受欢迎的微积分 >

integral of (tan^2(7x))/(csc(7x))

解答

\int \frac{tan ^{2} ( 7 x )}{csc ( 7 x )} d x

解答

\frac{1}{7} (sec (7 x) + cos (7 x)) + C

求解步骤

\int \frac{tan ^{2} ( 7 x )}{csc ( 7 x )} d x

使用换元积分法

= \int \frac{tan ^{2} ( u )}{7 csc ( u )} d u

提出常数:

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

= \frac{1}{7} \cdot \int \frac{tan ^{2} ( u )}{csc ( u )} d u

使用基本三角恒等式:

\frac{1}{csc ( x )} = sin (x)

= \frac{1}{7} \cdot \int tan^{2} (u) sin (u) d u

使用三角恒等式改写

= \frac{1}{7} \cdot \int \frac{( 1 - cos ^{2} ( u ) ) sin ( u )}{cos ^{2} ( u )} d u

使用换元积分法

= \frac{1}{7} \cdot \int - \frac{1 - v ^{2}}{v ^{2}} d v

乘开

- \frac{1 - v ^{2}}{v ^{2}} : - \frac{1}{v ^{2}} + 1

= \frac{1}{7} \cdot \int - \frac{1}{v ^{2}} + 1 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}{7} (- \int \frac{1}{v ^{2}} d v + \int 1 d v)

\int \frac{1}{v ^{2}} d v = - \frac{1}{v}

\int 1 d v = v

= \frac{1}{7} (- (- \frac{1}{v}) + v)

代回

= \frac{1}{7} (- (- \frac{1}{cos ( 7 x )}) + cos (7 x))

化简

\frac{1}{7} (- (- \frac{1}{cos ( 7 x )}) + cos (7 x)) : \frac{1}{7} (sec (7 x) + cos (7 x))

= \frac{1}{7} (sec (7 x) + cos (7 x))

解答补常数

= \frac{1}{7} (sec (7 x) + cos (7 x)) + C

作图

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