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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{24 ( 3 x - 4 ) ^{6}} + C

求解步骤

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

提出常数:

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

= \frac{3}{4} \cdot \int \frac{1}{( 4 - 3 x ) ^{7}} d x

将

\frac{1}{( 4 - 3 x ) ^{7}}

用部份分式展开:

- \frac{1}{( 3 x - 4 ) ^{7}}

= \frac{3}{4} \cdot \int - \frac{1}{( 3 x - 4 ) ^{7}} d x

提出常数:

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

= \frac{3}{4} (- \int \frac{1}{( 3 x - 4 ) ^{7}} d x)

使用换元积分法

= \frac{3}{4} (- \int \frac{1}{3 u ^{7}} d u)

提出常数:

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

= \frac{3}{4} (- \frac{1}{3} \cdot \int \frac{1}{u ^{7}} d u)

使用幂法则

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

u = 3 x - 4

代回

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

化简

\frac{3}{4} (- \frac{1}{3} (- \frac{1}{6 ( 3 x - 4 ) ^{6}})) : \frac{1}{24 ( 3 x - 4 ) ^{6}}

= \frac{1}{24 ( 3 x - 4 ) ^{6}}

解答补常数

= \frac{1}{24 ( 3 x - 4 ) ^{6}} + C

作图

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