What is the integral of xsqrt(4+x^2) ?

The integral of xsqrt(4+x^2) is 1/3 (4+x^2)^{3/2}+C

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integral of xsqrt(4+x^2)

\int x 4 + x^{2} d x

\frac{1}{3} (4 + x^{2})^{\frac{3}{2}} + C

Solution steps

\int x 4 + 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 = 4 + x^{2}

= \frac{( 4 + x ^{2} ) ^{3}}{3}

Simplify

\frac{( 4 + x ^{2} ) ^{3}}{3} : \frac{1}{3} (4 + x^{2})^{\frac{3}{2}}

= \frac{1}{3} (4 + x^{2})^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3} (4 + x^{2})^{\frac{3}{2}} + C

x + \frac{4}{x}

\frac{d}{d y} (\frac{y ^{4}}{4} + \frac{1}{8 y ^{2}})

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

\int 2 x^{\frac{1}{4}} - 3 x^{3} d x

\frac{d}{d θ} (14 sin (θ))

What is the integral of xsqrt(4+x^2) ?
The integral of xsqrt(4+x^2) is 1/3 (4+x^2)^{3/2}+C