What is the integral of 3x^6sqrt(2-x^7) ?

The integral of 3x^6sqrt(2-x^7) is -2/7 (2-x^7)^{3/2}+C

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integral of 3x^6sqrt(2-x^7)

\int 3 x^{6} 2 - x^{7} d x

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

Solution steps

\int 3 x^{6} 2 - x^{7} d x

Take the constant out:

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

= 3 \cdot \int x^{6} 2 - x^{7} d x

Apply u-substitution

: \int - \frac{u}{7} d u

= 3 \cdot \int - \frac{u}{7} d u

Take the constant out:

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

= 3 (- \frac{1}{7} \cdot \int u d u)

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

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

Substitute back

u = 2 - x^{7}

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 3x^6sqrt(2-x^7) ?
The integral of 3x^6sqrt(2-x^7) is -2/7 (2-x^7)^{3/2}+C