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

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

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integral of sqrt(1+r^2)r

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

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

Solution steps

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

Apply u-substitution

= \int u^{2} d u

Apply the Power Rule

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

Substitute back

u = 1 + r^{2}

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

Simplify

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

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

Add a constant to the solution

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

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