What is the integral of 2x^6(5-3x^7)^{1/4} ?

The integral of 2x^6(5-3x^7)^{1/4} is -8/105 (5-3x^7)^{5/4}+C

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integral of 2x^6(5-3x^7)^{1/4}

\int 2 x^{6} (5 - 3 x^{7})^{\frac{1}{4}} d x

- \frac{8}{105} (5 - 3 x^{7})^{\frac{5}{4}} + C

Solution steps

\int 2 x^{6} (5 - 3 x^{7})^{\frac{1}{4}} d x

Take the constant out:

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

= 2 \cdot \int x^{6} (5 - 3 x^{7})^{\frac{1}{4}} d x

Apply u-substitution

: \int - \frac{u ^{\frac{1}{4}}}{21} d u

= 2 \cdot \int - \frac{u ^{\frac{1}{4}}}{21} d u

Take the constant out:

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

= 2 (- \frac{1}{21} \cdot \int u^{\frac{1}{4}} d u)

Apply the Power Rule

: \frac{4}{5} u^{\frac{5}{4}}

= 2 (- \frac{1}{21} \cdot \frac{4}{5} u^{\frac{5}{4}})

Substitute back

u = 5 - 3 x^{7}

= 2 (- \frac{1}{21} \cdot \frac{4}{5} (5 - 3 x^{7})^{\frac{5}{4}})

Simplify

2 (- \frac{1}{21} \cdot \frac{4}{5} (5 - 3 x^{7})^{\frac{5}{4}}) : - \frac{8}{105} (5 - 3 x^{7})^{\frac{5}{4}}

= - \frac{8}{105} (5 - 3 x^{7})^{\frac{5}{4}}

Add a constant to the solution

= - \frac{8}{105} (5 - 3 x^{7})^{\frac{5}{4}} + C

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What is the integral of 2x^6(5-3x^7)^{1/4} ?
The integral of 2x^6(5-3x^7)^{1/4} is -8/105 (5-3x^7)^{5/4}+C