What is the integral of x^6sqrt(x^7+9) ?

The integral of x^6sqrt(x^7+9) is 2/21 (x^7+9)^{3/2}+C

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integral of x^6sqrt(x^7+9)

\int x^{6} x^{7} + 9 d x

\frac{2}{21} (x^{7} + 9)^{\frac{3}{2}} + C

Solution steps

\int x^{6} x^{7} + 9 d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = x^{7} + 9

= \frac{1}{7} \cdot \frac{2}{3} (x^{7} + 9)^{\frac{3}{2}}

Simplify

\frac{1}{7} \cdot \frac{2}{3} (x^{7} + 9)^{\frac{3}{2}} : \frac{2}{21} (x^{7} + 9)^{\frac{3}{2}}

= \frac{2}{21} (x^{7} + 9)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{21} (x^{7} + 9)^{\frac{3}{2}} + C

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What is the integral of x^6sqrt(x^7+9) ?
The integral of x^6sqrt(x^7+9) is 2/21 (x^7+9)^{3/2}+C