What is the integral of 5x^2sqrt((10x^3-7)^5) ?

The integral of 5x^2sqrt((10x^3-7)^5) is 1/21 (10x^3-7)^{7/2}+C

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integral of 5x^2sqrt((10x^3-7)^5)

\int 5 x^{2} (10 x^{3} - 7)^{5} d x

\frac{1}{21} (10 x^{3} - 7)^{\frac{7}{2}} + C

Solution steps

\int 5 x^{2} (10 x^{3} - 7)^{5} d x

Take the constant out:

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

= 5 \cdot \int x^{2} (10 x^{3} - 7)^{5} d x

Apply u-substitution

: \int \frac{u ^{\frac{5}{2}}}{30} d u

= 5 \cdot \int \frac{u ^{\frac{5}{2}}}{30} d u

Take the constant out:

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

= 5 \cdot \frac{1}{30} \cdot \int u^{\frac{5}{2}} d u

Apply the Power Rule

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

= 5 \cdot \frac{1}{30} \cdot \frac{2}{7} u^{\frac{7}{2}}

Substitute back

u = 10 x^{3} - 7

= 5 \cdot \frac{1}{30} \cdot \frac{2}{7} (10 x^{3} - 7)^{\frac{7}{2}}

Simplify

5 \cdot \frac{1}{30} \cdot \frac{2}{7} (10 x^{3} - 7)^{\frac{7}{2}} : \frac{1}{21} (10 x^{3} - 7)^{\frac{7}{2}}

= \frac{1}{21} (10 x^{3} - 7)^{\frac{7}{2}}

Add a constant to the solution

= \frac{1}{21} (10 x^{3} - 7)^{\frac{7}{2}} + C

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What is the integral of 5x^2sqrt((10x^3-7)^5) ?
The integral of 5x^2sqrt((10x^3-7)^5) is 1/21 (10x^3-7)^{7/2}+C