zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

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

解答

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

解答

\frac{1}{2} sec^{2} (x) + ln ∣ cos (x) ∣ + C

求解步骤

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

使用三角恒等式改写

= \int \frac{( 1 - cos ^{2} ( x ) ) sin ( x )}{cos ^{3} ( x )} d x

使用换元积分法

= \int - \frac{1 - u ^{2}}{u ^{3}} d u

乘开

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

= \int - \frac{1}{u ^{3}} + \frac{1}{u} 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 ^{3}} d u + \int \frac{1}{u} d u

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

\int \frac{1}{u} d u = ln ∣ u ∣

= - (- \frac{1}{2 u ^{2}}) + ln ∣ u ∣

u = cos (x)

代回

= - (- \frac{1}{2 cos ^{2} ( x )}) + ln ∣ cos (x) ∣

化简

- (- \frac{1}{2 cos ^{2} ( x )}) + ln ∣ cos (x) ∣ : \frac{1}{2} sec^{2} (x) + ln ∣ cos (x) ∣

= \frac{1}{2} sec^{2} (x) + ln ∣ cos (x) ∣

解答补常数

= \frac{1}{2} sec^{2} (x) + ln ∣ cos (x) ∣ + C

作图

流行的例子

derivative of 8x^{1/2}

derivative of 15e^{1/x}

limit as x approaches 0+of xln(tan(9x))

integral from-2pi to pi of cos^2(x)