What is the integral of x^2sqrt(x^3+15) ?

The integral of x^2sqrt(x^3+15) is 2/9 (x^3+15)^{3/2}+C

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integral of x^2sqrt(x^3+15)

\int x^{2} x^{3} + 15 d x

\frac{2}{9} (x^{3} + 15)^{\frac{3}{2}} + C

Solution steps

\int x^{2} x^{3} + 15 d x

Apply u-substitution

: \int \frac{u}{3} d u

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = x^{3} + 15

= \frac{1}{3} \cdot \frac{2}{3} (x^{3} + 15)^{\frac{3}{2}}

Simplify

\frac{1}{3} \cdot \frac{2}{3} (x^{3} + 15)^{\frac{3}{2}} : \frac{2}{9} (x^{3} + 15)^{\frac{3}{2}}

= \frac{2}{9} (x^{3} + 15)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{9} (x^{3} + 15)^{\frac{3}{2}} + C

n = 1 \sum \infty e^{(- n^{2})}

x \to - 5 lim (- \frac{9}{x + 5})

\frac{d}{d x} (ln (\frac{x ^{2} - 1}{x ^{2} + 1}))

\int (5 + x) d x

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

What is the integral of x^2sqrt(x^3+15) ?
The integral of x^2sqrt(x^3+15) is 2/9 (x^3+15)^{3/2}+C