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 tcsc^2((3t^2)/2)

解答

\int t csc^{2} (\frac{3 t ^{2}}{2}) d t

解答

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

求解步骤

\int t csc^{2} (\frac{3 t ^{2}}{2}) d t

使用换元积分法

= \int \frac{csc ^{2} ( \frac{3 u}{2} )}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int csc^{2} (\frac{3 u}{2}) d u

使用三角恒等式改写

= \frac{1}{2} \cdot \int \frac{1}{sin ^{2} ( \frac{3 u}{2} )} d u

使用换元积分法

= \frac{1}{2} \cdot \int \frac{2}{3 sin ^{2} ( v )} d v

提出常数:

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

= \frac{1}{2} \cdot \frac{2}{3} \cdot \int \frac{1}{sin ^{2} ( v )} d v

使用积分常见定则:

\int \frac{1}{sin ^{2} ( v )} d v = - cot (v)

= \frac{1}{2} \cdot \frac{2}{3} (- cot (v))

代回

= \frac{1}{2} \cdot \frac{2}{3} (- cot (\frac{3 t ^{2}}{2}))

化简

\frac{1}{2} \cdot \frac{2}{3} (- cot (\frac{3 t ^{2}}{2})) : - \frac{1}{3} cot (\frac{3 t ^{2}}{2})

= - \frac{1}{3} cot (\frac{3 t ^{2}}{2})

解答补常数

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

作图

流行的例子

sum from n=4 to infinity of 1/(n^2)

n = 4 \sum \infty \frac{1}{n ^{2}}

tangent f(x)= 6/(5x+2),(-1,-2)

t an g e n t f (x) = \frac{6}{5 x + 2}, (- 1, - 2)

integral of ((sin(4x))/4)-((sin^3(x))/4)

\int (\frac{sin ( 4 x )}{4}) - (\frac{sin ^{3} ( x )}{4}) d x

sum from n=0 to infinity of n(n+1)-1

n = 0 \sum \infty n (n + 1) - 1

integral of cos(x+pi)

\int cos (x + π) d x