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

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

integral of xsqrt(3x^2+2)

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

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

Solution steps

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

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

Simplify

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

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

Add a constant to the solution

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

\int (cos (2 x) - sin (2 x))^{2} d x

\int x (6 x^{2} + x^{\frac{1}{2}}) d x

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

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

\int ln (x^{2} + 6 x + 10) d x

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