What is the integral of x^2sqrt(x^3+25) ?

The integral of x^2sqrt(x^3+25) is 2/9 (x^3+25)^{3/2}+C

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integral of x^2sqrt(x^3+25)

\int x^{2} x^{3} + 25 d x

\frac{2}{9} (x^{3} + 25)^{\frac{3}{2}} + C

Solution steps

\int x^{2} x^{3} + 25 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} + 25

= \frac{1}{3} \cdot \frac{2}{3} (x^{3} + 25)^{\frac{3}{2}}

Simplify

\frac{1}{3} \cdot \frac{2}{3} (x^{3} + 25)^{\frac{3}{2}} : \frac{2}{9} (x^{3} + 25)^{\frac{3}{2}}

= \frac{2}{9} (x^{3} + 25)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{9} (x^{3} + 25)^{\frac{3}{2}} + C

\int \frac{x}{3 - x} d x

\int e^{x} (\frac{1}{x} + ln (x)) d x

\int \frac{1}{( x ^{2} + 4 x + 7 ) ^{3}} d x

\int (e^{\frac{x}{2}} - x) (e^{\frac{x}{2}} + x) d x

\frac{1}{1 + x ^{2}}

What is the integral of x^2sqrt(x^3+25) ?
The integral of x^2sqrt(x^3+25) is 2/9 (x^3+25)^{3/2}+C