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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

: \int cos (u) d u

= \int cos (u) d u

Use the common integral:

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

= sin (u)

Substitute back

u = arctan (y)

= sin (arctan (y))

Use the following identity:

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

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

Add a constant to the solution

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

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