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

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

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integral of 1/((x^2+5)^{3/2)}

\int \frac{1}{( x ^{2} + 5 ) ^{\frac{3}{2}}} d x

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

Solution steps

\int \frac{1}{( x ^{2} + 5 ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

: \int \frac{1}{5 sec ( u )} d u

= \int \frac{1}{5 sec ( u )} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{5} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{5} sin (u)

Substitute back

u = arctan (\frac{1}{5 ^{\frac{1}{2}}} x)

= \frac{1}{5} sin (arctan (\frac{1}{5 ^{\frac{1}{2}}} x))

Simplify

\frac{1}{5} sin (arctan (\frac{1}{5 ^{\frac{1}{2}}} x)) : \frac{x}{5 ( 5 + x ^{2} ) ^{\frac{1}{2}}}

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (cos (n x) e^{n y})

\int \frac{arctan ( 5 x )}{1 + 25 x ^{2}} d x

f (x) = \frac{x ^{4} + 5 x ^{2} + 10}{x ^{2}}

\int e^{x} - e^{- x} d x

x = 1 - y^{2}, x = y^{2} - 1

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