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 from 0 to pi/4 of 7tan^3(x)

解答

\int_{0}^{\frac{π}{4}} 7 tan^{3} (x) d x

解答

7 (- \frac{1}{2} ln (2) + \frac{1}{2})

+1

十进制

1.07398 \dots

求解步骤

\int_{0}^{\frac{π}{4}} 7 tan^{3} (x) d x

提出常数:

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

= 7 \cdot \int_{0}^{\frac{π}{4}} tan^{3} (x) d x

使用三角恒等式改写

= 7 \cdot \int_{0}^{\frac{π}{4}} (- 1 + sec^{2} (x)) tan (x) d x

使用换元积分法

= 7 \cdot \int_{1}^{2} \frac{- 1 + u ^{2}}{u} d u

乘开

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

= 7 \cdot \int_{1}^{2} - \frac{1}{u} + 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

= 7 (- \int_{1}^{2} \frac{1}{u} d u + \int_{1}^{2} u d u)

\int_{1}^{2} \frac{1}{u} d u = \frac{1}{2} ln (2)

\int_{1}^{2} u d u = \frac{1}{2}

= 7 (- \frac{1}{2} ln (2) + \frac{1}{2})

作图

流行的例子

integral of (3x)/((x^2+3)^2)

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

derivative f(x)=sqrt(x)+2

d er i v a t i v e f (x) = x + 2

limit as x approaches-1 of x^2-3x+1

x \to - 1 lim (x^{2} - 3 x + 1)

derivative of (sqrt(3x-1)/((2-x)^2))

\frac{d}{d x} (\frac{3 x - 1}{( 2 - x ) ^{2}})

d x - 2 x y d y = 0