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受欢迎的微积分 >

integral of (x^7)/(sqrt(x^4-3))

解答

\int \frac{x ^{7}}{x ^{4} - 3} d x

解答

\frac{1}{6} (x^{4} + 6) x^{4} - 3 + C

求解步骤

\int \frac{x ^{7}}{x ^{4} - 3} d x

使用换元积分法

= \int \frac{u + 3}{4 u} d u

提出常数:

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

= \frac{1}{4} \cdot \int \frac{u + 3}{u} d u

乘开

\frac{u + 3}{u} : u + \frac{3}{u}

= \frac{1}{4} \cdot \int u + \frac{3}{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

= \frac{1}{4} (\int u d u + \int \frac{3}{u} d u)

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

\int \frac{3}{u} d u = 6 u

= \frac{1}{4} (\frac{2}{3} u^{\frac{3}{2}} + 6 u)

u = x^{4} - 3

代回

= \frac{1}{4} (\frac{2}{3} (x^{4} - 3)^{\frac{3}{2}} + 6 x^{4} - 3)

化简

\frac{1}{4} (\frac{2}{3} (x^{4} - 3)^{\frac{3}{2}} + 6 x^{4} - 3) : \frac{1}{6} (x^{4} + 6) x^{4} - 3

= \frac{1}{6} (x^{4} + 6) x^{4} - 3

解答补常数

= \frac{1}{6} (x^{4} + 6) x^{4} - 3 + C

作图

流行的例子

斜率 (-7,-4),(8,8)

s l o p e (- 7, - 4), (8, 8)

sum from n=1 to infinity of (6^n)/((4+11^n))

n = 1 \sum \infty \frac{6 ^{n}}{( 4 + 1 1 ^{n} )}

integral of 4/(x^2+25)

\int \frac{4}{x ^{2} + 25} d x

derivative (6x)/(e^x)

d er i v a t i v e \frac{6 x}{e ^{x}}

xy^'+6y=-35xcos(x^7)

x y^{^{'}} + 6 y = - 35 x cos (x^{7})