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

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

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

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

- \frac{1}{3 ( x ^{2} + 1 ) ^{\frac{3}{2}}} + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u ^{\frac{5}{2}}} d u

= \int \frac{1}{2 u ^{\frac{5}{2}}} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{\frac{5}{2}}} d u

Apply the Power Rule

: - \frac{2}{3 u ^{\frac{3}{2}}}

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

Substitute back

u = x^{2} + 1

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{3 ( x ^{2} + 1 ) ^{\frac{3}{2}}} + C

\int \frac{y ^{3}}{y ^{4} + 1} d y

(x^{2} + 1) y^{^{'}} + 3 x (y - 1) = 0, y (0) = 8

\int \frac{sin ( c x )}{cos ^{2} ( c x )} d x

\frac{14 x + 8 x ^{2}}{( 7 + 8 x ) ^{2}}

\frac{d}{d t} x + 7 x = 5 cos (2 t)

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