What is the integral of x/(sqrt((x^2+2x+10)^3)) ?

The integral of x/(sqrt((x^2+2x+10)^3)) is (x(x+1))/(9sqrt(x^2+2x+10))-1/9 sqrt(x^2+2x+10)+C

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integral of x/(sqrt((x^2+2x+10)^3))

\int \frac{x}{( x ^{2} + 2 x + 10 ) ^{3}} d x

\frac{x ( x + 1 )}{9 x ^{2} + 2 x + 10} - \frac{1}{9} x^{2} + 2 x + 10 + C

Solution steps

\int \frac{x}{( x ^{2} + 2 x + 10 ) ^{3}} d x

Apply Integration By Parts

: \frac{x ( x + 1 )}{9 x ^{2} + 2 x + 10} - \int \frac{x + 1}{9 x ^{2} + 2 x + 10} d x

= \frac{x ( x + 1 )}{9 x ^{2} + 2 x + 10} - \int \frac{x + 1}{9 x ^{2} + 2 x + 10} d x

\int \frac{x + 1}{9 x ^{2} + 2 x + 10} d x = \frac{1}{9} x^{2} + 2 x + 10

= \frac{x ( x + 1 )}{9 x ^{2} + 2 x + 10} - \frac{1}{9} x^{2} + 2 x + 10

Add a constant to the solution

= \frac{x ( x + 1 )}{9 x ^{2} + 2 x + 10} - \frac{1}{9} x^{2} + 2 x + 10 + C

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What is the integral of x/(sqrt((x^2+2x+10)^3)) ?
The integral of x/(sqrt((x^2+2x+10)^3)) is (x(x+1))/(9sqrt(x^2+2x+10))-1/9 sqrt(x^2+2x+10)+C