What is the integral of 1/((x^2+2x+2)^{3/2)} ?

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

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integral of 1/((x^2+2x+2)^{3/2)}

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

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

Solution steps

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

Complete the square

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

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

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int cos (v) d v

= \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= sin (v)

Substitute back

= sin (arctan (x + 1))

Simplify

sin (arctan (x + 1)) : \frac{x + 1}{( x ^{2} + 2 x + 2 ) ^{\frac{1}{2}}}

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

Add a constant to the solution

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

\int \frac{1}{x ^{\frac{1}{n}}} d x

\int \frac{sin ( 11 x )}{1 + cos ^{2} ( 11 x )} d x

\int \frac{x}{4 x ^{2} - 1} d x

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

\int (x + 2) x - 1 d x

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