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受欢迎的微积分 >

integral of x/(sqrt(1-4x))

解答

\int \frac{x}{1 - 4 x} d x

解答

- \frac{1}{12} (2 x + 1) - 4 x + 1 + C

求解步骤

\int \frac{x}{1 - 4 x} d x

使用换元积分法

= \int \frac{u ^{2} - 1}{8} d u

提出常数:

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

= \frac{1}{8} \cdot \int u^{2} - 1 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}{8} (\int u^{2} d u - \int 1 d u)

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

\int 1 d u = u

= \frac{1}{8} (\frac{u ^{3}}{3} - u)

u = 1 - 4 x

代回

= \frac{1}{8} (\frac{( 1 - 4 x ) ^{3}}{3} - 1 - 4 x)

化简

\frac{1}{8} (\frac{( 1 - 4 x ) ^{3}}{3} - 1 - 4 x) : - \frac{1}{12} (2 x + 1) - 4 x + 1

= - \frac{1}{12} (2 x + 1) - 4 x + 1

解答补常数

= - \frac{1}{12} (2 x + 1) - 4 x + 1 + C

作图

流行的例子

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

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

integral of (x^2-5)/((1+x)^3)

\int \frac{x ^{2} - 5}{( 1 + x ) ^{3}} d x

in t e g r a l \frac{1}{4 x}

integral of 6x+12x^2

\int 6 x + 12 x^{2} d x

integral of (cot^3(x))/4

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