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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

= \int \frac{1}{a ^{2} sec ( u )} d u

Take the constant out:

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

= \frac{1}{a ^{2}} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{a ^{2}} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{a ^{2}} sin (u)

Substitute back

u = arctan (\frac{1}{a} y)

= \frac{1}{a ^{2}} sin (arctan (\frac{1}{a} y))

Simplify

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

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

Add a constant to the solution

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

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