What is the integral of x^5sqrt(6x^6+2) ?

The integral of x^5sqrt(6x^6+2) is 1/54 (6x^6+2)^{3/2}+C

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integral of x^5sqrt(6x^6+2)

\int x^{5} 6 x^{6} + 2 d x

\frac{1}{54} (6 x^{6} + 2)^{\frac{3}{2}} + C

Solution steps

\int x^{5} 6 x^{6} + 2 d x

Apply u-substitution

: \int \frac{u}{36} d u

= \int \frac{u}{36} d u

Take the constant out:

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

= \frac{1}{36} \cdot \int u d u

Apply the Power Rule

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

= \frac{1}{36} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 6 x^{6} + 2

= \frac{1}{36} \cdot \frac{2}{3} (6 x^{6} + 2)^{\frac{3}{2}}

Simplify

\frac{1}{36} \cdot \frac{2}{3} (6 x^{6} + 2)^{\frac{3}{2}} : \frac{1}{54} (6 x^{6} + 2)^{\frac{3}{2}}

= \frac{1}{54} (6 x^{6} + 2)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{54} (6 x^{6} + 2)^{\frac{3}{2}} + C

\int (4 x^{2} + 3)^{2} d x

x^{2} - x, 2 x - 2 x^{2}

x \to 4 lim (\frac{1}{x + 2})

e^{x} - 6 x^{5}

\int (3 x^{2} - 4)^{2} x^{3} d x

What is the integral of x^5sqrt(6x^6+2) ?
The integral of x^5sqrt(6x^6+2) is 1/54 (6x^6+2)^{3/2}+C