What is the integral of x/((x^2+4x+5)^{3/2)} ?

The integral of x/((x^2+4x+5)^{3/2)} is -1/((x^2+4x+5)^{1/2)}-(2(x+2))/((x^2+4x+5)^{1/2)}+C

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integral of x/((x^2+4x+5)^{3/2)}

\int \frac{x}{( x ^{2} + 4 x + 5 ) ^{\frac{3}{2}}} d x

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

Solution steps

\int \frac{x}{( x ^{2} + 4 x + 5 ) ^{\frac{3}{2}}} d x

Complete the square

x^{2} + 4 x + 5 : (x + 2)^{2} + 1

= \int \frac{x}{( ( x + 2 ) ^{2} + 1 ) ^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{u - 2}{( u ^{2} + 1 ) ^{\frac{3}{2}}} d u

= \int \frac{u - 2}{( u ^{2} + 1 ) ^{\frac{3}{2}}} d u

Expand

\frac{u - 2}{( u ^{2} + 1 ) ^{\frac{3}{2}}} : \frac{u}{( u ^{2} + 1 ) ^{\frac{3}{2}}} - \frac{2}{( u ^{2} + 1 ) ^{\frac{3}{2}}}

= \int \frac{u}{( u ^{2} + 1 ) ^{\frac{3}{2}}} - \frac{2}{( u ^{2} + 1 ) ^{\frac{3}{2}}} d u

Apply the Sum Rule:

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

= \int \frac{u}{( u ^{2} + 1 ) ^{\frac{3}{2}}} d u - \int \frac{2}{( u ^{2} + 1 ) ^{\frac{3}{2}}} d u

\int \frac{u}{( u ^{2} + 1 ) ^{\frac{3}{2}}} d u = - \frac{1}{( u ^{2} + 1 ) ^{\frac{1}{2}}}

\int \frac{2}{( u ^{2} + 1 ) ^{\frac{3}{2}}} d u = \frac{2 u}{( 1 + u ^{2} ) ^{\frac{1}{2}}}

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

Substitute back

u = x + 2

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

Simplify

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

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

Add a constant to the solution

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

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