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受欢迎的微积分 >

integral of 1/(x^2sqrt(3x^2+4))

解答

\int \frac{1}{x ^{2} 3 x ^{2} + 4} d x

解答

- \frac{4 + 3 x ^{2}}{4 x} + C

求解步骤

\int \frac{1}{x ^{2} 3 x ^{2} + 4} d x

使用三角换元法

= \int \frac{3 sec ( u )}{4 tan ^{2} ( u )} d u

提出常数:

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

= \frac{3}{4} \cdot \int \frac{sec ( u )}{tan ^{2} ( u )} d u

使用三角恒等式改写

= \frac{3}{4} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

使用换元积分法

= \frac{3}{4} \cdot \int \frac{1}{v ^{2}} d v

使用幂法则

= \frac{3}{4} (- \frac{1}{v})

代回

= \frac{3}{4} - \frac{1}{sin ( arctan ( \frac{3}{2} x ) )}

化简

\frac{3}{4} - \frac{1}{sin ( arctan ( \frac{3}{2} x ) )} : - \frac{4 + 3 x ^{2}}{4 x}

= - \frac{4 + 3 x ^{2}}{4 x}

解答补常数

= - \frac{4 + 3 x ^{2}}{4 x} + C

作图

流行的例子

(\partial)/(\partial y)(x/(x^2+y^2+z^2))

\frac{\partial}{\partial y} (\frac{x}{x ^{2} + y ^{2} + z ^{2}})

integral from 1 to 5e of 1/(3x)

\int_{1}^{5 e} \frac{1}{3 x} d x

integral of (2-3x)^5

\int (2 - 3 x)^{5} d x

sum from n=5 to infinity of sin^2(n)

n = 5 \sum \infty sin^{2} (n)

integral of 4sin(3t)+1.2e^{-6t}

\int 4 sin (3 t) + 1.2 e^{- 6 t} d t