What is the integral of x^2sqrt((5x^3-3)^5) ?

The integral of x^2sqrt((5x^3-3)^5) is 2/105 (5x^3-3)^{7/2}+C

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integral of x^2sqrt((5x^3-3)^5)

\int x^{2} (5 x^{3} - 3)^{5} d x

\frac{2}{105} (5 x^{3} - 3)^{\frac{7}{2}} + C

Solution steps

\int x^{2} (5 x^{3} - 3)^{5} d x

Apply u-substitution

: \int \frac{u ^{\frac{5}{2}}}{15} d u

= \int \frac{u ^{\frac{5}{2}}}{15} d u

Take the constant out:

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

= \frac{1}{15} \cdot \int u^{\frac{5}{2}} d u

Apply the Power Rule

: \frac{2}{7} u^{\frac{7}{2}}

= \frac{1}{15} \cdot \frac{2}{7} u^{\frac{7}{2}}

Substitute back

u = 5 x^{3} - 3

= \frac{1}{15} \cdot \frac{2}{7} (5 x^{3} - 3)^{\frac{7}{2}}

Simplify

\frac{1}{15} \cdot \frac{2}{7} (5 x^{3} - 3)^{\frac{7}{2}} : \frac{2}{105} (5 x^{3} - 3)^{\frac{7}{2}}

= \frac{2}{105} (5 x^{3} - 3)^{\frac{7}{2}}

Add a constant to the solution

= \frac{2}{105} (5 x^{3} - 3)^{\frac{7}{2}} + C

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What is the integral of x^2sqrt((5x^3-3)^5) ?
The integral of x^2sqrt((5x^3-3)^5) is 2/105 (5x^3-3)^{7/2}+C