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受欢迎的微积分 >

integral of (sec^6(θ))/(tan^2(θ))

解答

\int \frac{sec ^{6} ( θ )}{tan ^{2} ( θ )} d θ

解答

- cot (θ) + 2 tan (θ) + \frac{1}{3} tan^{3} (θ) + C

求解步骤

\int \frac{sec ^{6} ( θ )}{tan ^{2} ( θ )} d θ

使用三角恒等式改写

= \int \frac{1}{cos ^{4} ( θ ) sin ^{2} ( θ )} d θ

使用三角换元法

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

乘开

\frac{( 1 + u ^{2} ) ^{2}}{u ^{2}} : \frac{1}{u ^{2}} + 2 + u^{2}

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

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

\int 2 d u = 2 u

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

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

u = tan (θ)

代回

= - \frac{1}{tan ( θ )} + 2 tan (θ) + \frac{tan ^{3} ( θ )}{3}

化简

- \frac{1}{tan ( θ )} + 2 tan (θ) + \frac{tan ^{3} ( θ )}{3} : - cot (θ) + 2 tan (θ) + \frac{1}{3} tan^{3} (θ)

= - cot (θ) + 2 tan (θ) + \frac{1}{3} tan^{3} (θ)

解答补常数

= - cot (θ) + 2 tan (θ) + \frac{1}{3} tan^{3} (θ) + C

作图

流行的例子

integral of a*e^{-ax}

\int a \cdot e^{- a x} d x

integral of arctan(tan(t))

\int arctan (tan (t)) d t

integral of (\sqrt[3]{x}-4+5x)/(x^2)

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

integral of xsqrt((1+x^2))

\int x (1 + x^{2}) d x

integral of (1-sin(x))/(2+cos(x))

\int \frac{1 - sin ( x )}{2 + cos ( x )} d x