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受欢迎的微积分 >

integral of tan^{-8}(x)sec^4(x)

解答

\int tan^{- 8} (x) sec^{4} (x) d x

解答

- \frac{1}{7} cot^{7} (x) - \frac{1}{5} cot^{5} (x) + C

求解步骤

\int tan^{- 8} (x) sec^{4} (x) d x

使用三角恒等式改写

= \int (1 + tan^{2} (x)) sec^{2} (x) tan^{- 8} (x) d x

使用换元积分法

= \int \frac{1 + u ^{2}}{u ^{8}} d u

乘开

\frac{1 + u ^{2}}{u ^{8}} : \frac{1}{u ^{8}} + \frac{1}{u ^{6}}

= \int \frac{1}{u ^{8}} + \frac{1}{u ^{6}} 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

= \int \frac{1}{u ^{8}} d u + \int \frac{1}{u ^{6}} d u

\int \frac{1}{u ^{8}} d u = - \frac{1}{7 u ^{7}}

\int \frac{1}{u ^{6}} d u = - \frac{1}{5 u ^{5}}

= - \frac{1}{7 u ^{7}} - \frac{1}{5 u ^{5}}

u = tan (x)

代回

= - \frac{1}{7 tan ^{7} ( x )} - \frac{1}{5 tan ^{5} ( x )}

化简

- \frac{1}{7 tan ^{7} ( x )} - \frac{1}{5 tan ^{5} ( x )} : - \frac{1}{7} cot^{7} (x) - \frac{1}{5} cot^{5} (x)

= - \frac{1}{7} cot^{7} (x) - \frac{1}{5} cot^{5} (x)

解答补常数

= - \frac{1}{7} cot^{7} (x) - \frac{1}{5} cot^{5} (x) + C

作图

流行的例子

integral of sin(3t)sec^2(cos(3t))

\int sin (3 t) sec^{2} (cos (3 t)) d t

integral of ((x+1)^3)/x

\int \frac{( x + 1 ) ^{3}}{x} d x

integral of (4cosh(x)+x^5)

\int (4 cosh (x) + x^{5}) d x

integral of sqrt(x^2+z^2)

\int x^{2} + z^{2} d z

integral of-5cos(pi)x

\int - 5 cos (π) x d x