What is the integral of 3xsqrt(3x^2+7) ?

The integral of 3xsqrt(3x^2+7) is 1/3 (3x^2+7)^{3/2}+C

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integral of 3xsqrt(3x^2+7)

\int 3 x 3 x^{2} + 7 d x

\frac{1}{3} (3 x^{2} + 7)^{\frac{3}{2}} + C

Solution steps

\int 3 x 3 x^{2} + 7 d x

Take the constant out:

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

= 3 \cdot \int x 3 x^{2} + 7 d x

Apply u-substitution

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

= 3 \cdot \int \frac{u}{6} d u

Take the constant out:

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

= 3 \cdot \frac{1}{6} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

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

Substitute back

u = 3 x^{2} + 7

= 3 \cdot \frac{1}{6} \cdot \frac{2}{3} (3 x^{2} + 7)^{\frac{3}{2}}

Simplify

3 \cdot \frac{1}{6} \cdot \frac{2}{3} (3 x^{2} + 7)^{\frac{3}{2}} : \frac{1}{3} (3 x^{2} + 7)^{\frac{3}{2}}

= \frac{1}{3} (3 x^{2} + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3} (3 x^{2} + 7)^{\frac{3}{2}} + C

x + 1

\int \frac{cot ( 3 + ln ( x ) )}{x} d x

\int 2 y + cos (y) d y

\int \frac{x ^{2} + 1}{x ^{3} + 1} d x

\int f (2 x) d x

What is the integral of 3xsqrt(3x^2+7) ?
The integral of 3xsqrt(3x^2+7) is 1/3 (3x^2+7)^{3/2}+C