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-pi/6 to 0 of tan(x)

解答

\int_{- \frac{π}{6}}^{0} tan (x) d x

解答

\frac{1}{2} ln (3) - ln (2)

+1

十进制

- 0.14384 \dots

求解步骤

\int_{- \frac{π}{6}}^{0} tan (x) d x

使用三角恒等式改写

= \int_{- \frac{π}{6}}^{0} \frac{sin ( x )}{cos ( x )} d x

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - [ln ∣ u ∣]_{\frac{3}{2}}^{1}

带入上下限计算后得:

- (\frac{1}{2} ln (3) - ln (2))

= - (- (\frac{1}{2} ln (3) - ln (2)))

化简

= \frac{1}{2} ln (3) - ln (2)

作图

流行的例子

integral from 4 to 7 of 3+2ln(x+5)

\int_{4}^{7} 3 + 2 ln (x + 5) d x

integral from 0 to 2pi of cos(x)sin^3(x)

\int_{0}^{2 π} cos (x) sin^{3} (x) d x

integral from 0 to x of sin(8t^2)

\int_{0}^{x} sin (8 t^{2}) d t

integral from 1 to 4 of 3x+sqrt(x)

\int_{1}^{4} 3 x + x d x

integral from 1 to infinity of e^{-st}(1)

\int_{1}^{\infty} e^{- s t} (1) d t