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

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

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integral of xsqrt(x^2+3)

\int x x^{2} + 3 d x

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

Solution steps

\int x x^{2} + 3 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 = x^{2} + 3

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{x}{- x ^{2} + 5} d x

\int_{0}^{\frac{3}{2}} x cos (π x) d x

- \frac{1}{2 x ^{2}}

x \to \infty lim (\frac{x ^{3} + 20 x}{10 x - 2})

\frac{\partial}{\partial x} (cos (x) sin (yz))

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