What is the integral of xsqrt(3x^2+5) ?

The integral of xsqrt(3x^2+5) is 1/9 (3x^2+5)^{3/2}+C

integral of xsqrt(3x^2+5)

\int x 3 x^{2} + 5 d x

\frac{1}{9} (3 x^{2} + 5)^{\frac{3}{2}} + C

Solution steps

\int x 3 x^{2} + 5 d x

Apply u-substitution

= \int \frac{u}{6} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{6} \cdot \int u d u

Apply the Power Rule

= \frac{1}{6} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 3 x^{2} + 5

= \frac{1}{6} \cdot \frac{2}{3} (3 x^{2} + 5)^{\frac{3}{2}}

Simplify

\frac{1}{6} \cdot \frac{2}{3} (3 x^{2} + 5)^{\frac{3}{2}} : \frac{1}{9} (3 x^{2} + 5)^{\frac{3}{2}}

= \frac{1}{9} (3 x^{2} + 5)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{9} (3 x^{2} + 5)^{\frac{3}{2}} + C

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

\int \frac{1}{9 - x ^{4}} d x

\int x^{3} x^{2} + 14 d x

\int \frac{x - cos ( x )}{1 + sin ( x )} d x

\int x^{2} 3 x^{3} + 5 d x

What is the integral of xsqrt(3x^2+5) ?
The integral of xsqrt(3x^2+5) is 1/9 (3x^2+5)^{3/2}+C