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

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

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integral of x(sqrt(1+x^2))

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

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

Solution steps

\int x 1 + x^{2} 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

\frac{d}{d x} (\frac{3 x ^{2}}{y ^{2}})

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