What is the integral of xsqrt((1+x^2)^3) ?

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

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integral of xsqrt((1+x^2)^3)

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

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

Solution steps

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

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

assuming

(1 + x^{2}) \geq 0

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

Apply u-substitution

= \int \frac{u ^{\frac{3}{2}}}{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 u^{\frac{3}{2}} d u

Apply the Power Rule

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

Substitute back

u = 1 + x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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