What is the integral of (x^3-2)^{1/7}x^2 ?

The integral of (x^3-2)^{1/7}x^2 is 7/24 (x^3-2)^{8/7}+C

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integral of (x^3-2)^{1/7}x^2

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

\frac{7}{24} (x^{3} - 2)^{\frac{8}{7}} + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{\frac{1}{7}}}{3} d u

= \int \frac{u ^{\frac{1}{7}}}{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^{\frac{1}{7}} d u

Apply the Power Rule

: \frac{7}{8} u^{\frac{8}{7}}

= \frac{1}{3} \cdot \frac{7}{8} u^{\frac{8}{7}}

Substitute back

u = x^{3} - 2

= \frac{1}{3} \cdot \frac{7}{8} (x^{3} - 2)^{\frac{8}{7}}

Simplify

\frac{1}{3} \cdot \frac{7}{8} (x^{3} - 2)^{\frac{8}{7}} : \frac{7}{24} (x^{3} - 2)^{\frac{8}{7}}

= \frac{7}{24} (x^{3} - 2)^{\frac{8}{7}}

Add a constant to the solution

= \frac{7}{24} (x^{3} - 2)^{\frac{8}{7}} + C

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What is the integral of (x^3-2)^{1/7}x^2 ?
The integral of (x^3-2)^{1/7}x^2 is 7/24 (x^3-2)^{8/7}+C