What is the integral of 1/((2+x)^5) ?

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

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

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

\frac{x}{( 2 + x ) ^{5}} - \frac{5}{4 ( 2 + x ) ^{4}} + \frac{2}{( 2 + x ) ^{5}} + C

Solution steps

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

Apply Integration By Parts

: \frac{x}{( 2 + x ) ^{5}} - \int - \frac{5 x}{( 2 + x ) ^{6}} d x

= \frac{x}{( 2 + x ) ^{5}} - \int - \frac{5 x}{( 2 + x ) ^{6}} d x

\int - \frac{5 x}{( 2 + x ) ^{6}} d x = \frac{5}{4 ( 2 + x ) ^{4}} - \frac{2}{( 2 + x ) ^{5}}

= \frac{x}{( 2 + x ) ^{5}} - (\frac{5}{4 ( 2 + x ) ^{4}} - \frac{2}{( 2 + x ) ^{5}})

Simplify

= \frac{x}{( 2 + x ) ^{5}} - \frac{5}{4 ( 2 + x ) ^{4}} + \frac{2}{( 2 + x ) ^{5}}

Add a constant to the solution

= \frac{x}{( 2 + x ) ^{5}} - \frac{5}{4 ( 2 + x ) ^{4}} + \frac{2}{( 2 + x ) ^{5}} + C

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