What is the integral of (8x)/((5-2x)^5) ?

The integral of (8x)/((5-2x)^5) is 8(1/(12(2x-5)^3)+5/(16(2x-5)^4))+C

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integral of (8x)/((5-2x)^5)

\int \frac{8 x}{( 5 - 2 x ) ^{5}} d x

8 (\frac{1}{12 ( 2 x - 5 ) ^{3}} + \frac{5}{16 ( 2 x - 5 ) ^{4}}) + C

Solution steps

\int \frac{8 x}{( 5 - 2 x ) ^{5}} d x

Take the constant out:

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

= 8 \cdot \int \frac{x}{( 5 - 2 x ) ^{5}} d x

Take the partial fraction of

\frac{x}{( 5 - 2 x ) ^{5}} : - \frac{1}{2 ( 2 x - 5 ) ^{4}} - \frac{5}{2 ( 2 x - 5 ) ^{5}}

= 8 \cdot \int - \frac{1}{2 ( 2 x - 5 ) ^{4}} - \frac{5}{2 ( 2 x - 5 ) ^{5}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 8 (- \int \frac{1}{2 ( 2 x - 5 ) ^{4}} d x - \int \frac{5}{2 ( 2 x - 5 ) ^{5}} d x)

\int \frac{1}{2 ( 2 x - 5 ) ^{4}} d x = - \frac{1}{12 ( 2 x - 5 ) ^{3}}

\int \frac{5}{2 ( 2 x - 5 ) ^{5}} d x = - \frac{5}{16 ( 2 x - 5 ) ^{4}}

= 8 (- (- \frac{1}{12 ( 2 x - 5 ) ^{3}}) - (- \frac{5}{16 ( 2 x - 5 ) ^{4}}))

Simplify

= 8 (\frac{1}{12 ( 2 x - 5 ) ^{3}} + \frac{5}{16 ( 2 x - 5 ) ^{4}})

Add a constant to the solution

= 8 (\frac{1}{12 ( 2 x - 5 ) ^{3}} + \frac{5}{16 ( 2 x - 5 ) ^{4}}) + C

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What is the integral of (8x)/((5-2x)^5) ?
The integral of (8x)/((5-2x)^5) is 8(1/(12(2x-5)^3)+5/(16(2x-5)^4))+C