What is the integral of (2x)/((x+y)^3) ?

The integral of (2x)/((x+y)^3) is -x/((y+x)^2)-1/(y+x)+C

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integral of (2x)/((x+y)^3)

integral

\frac{2 x}{( x + y ) ^{3}}

- \frac{x}{( y + x ) ^{2}} - \frac{1}{y + x} + C

Solution steps

\int \frac{2 x}{( x + y ) ^{3}} d x

Take the constant out:

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

= 2 \cdot \int \frac{x}{( y + x ) ^{3}} d x

Apply Integration By Parts

: - \frac{x}{2 ( y + x ) ^{2}} - \int - \frac{1}{2 ( y + x ) ^{2}} d x

= 2 (- \frac{x}{2 ( y + x ) ^{2}} - \int - \frac{1}{2 ( y + x ) ^{2}} d x)

\int - \frac{1}{2 ( y + x ) ^{2}} d x = \frac{1}{2 ( y + x )}

= 2 (- \frac{x}{2 ( y + x ) ^{2}} - \frac{1}{2 ( y + x )})

Simplify

2 (- \frac{x}{2 ( y + x ) ^{2}} - \frac{1}{2 ( y + x )}) : - \frac{x}{( y + x ) ^{2}} - \frac{1}{y + x}

= - \frac{x}{( y + x ) ^{2}} - \frac{1}{y + x}

Add a constant to the solution

= - \frac{x}{( y + x ) ^{2}} - \frac{1}{y + x} + C

\frac{2}{( x + 1 ) ^{2}}

e^{- x} cos^{2} (x)

\frac{20000}{( x + 100 ) ^{3}}

\frac{( x ^{3} - x - 2 )}{( 1 - x ^{2} )}

x^{3} - 3 x^{2} + 5 x + 2

What is the integral of (2x)/((x+y)^3) ?
The integral of (2x)/((x+y)^3) is -x/((y+x)^2)-1/(y+x)+C