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

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

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integral of sqrt(1+x^2)x

\int 1 + x^{2} x d x

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

Solution steps

\int 1 + x^{2} x d x

Apply u-substitution

: \int u^{2} d u

= \int u^{2} d u

Apply the Power Rule

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

= \frac{u ^{3}}{3}

Substitute back

u = 1 + x^{2}

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

Simplify

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

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

Add a constant to the solution

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

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