What is the integral of 7x^2(x^3-8)^{19} ?

The integral of 7x^2(x^3-8)^{19} is 7/60 (x^3-8)^{20}+C

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integral of 7x^2(x^3-8)^{19}

\int 7 x^{2} (x^{3} - 8)^{19} d x

\frac{7}{60} (x^{3} - 8)^{20} + C

Solution steps

\int 7 x^{2} (x^{3} - 8)^{19} d x

Take the constant out:

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

= 7 \cdot \int x^{2} (x^{3} - 8)^{19} d x

Apply u-substitution

: \int \frac{u ^{19}}{3} d u

= 7 \cdot \int \frac{u ^{19}}{3} d u

Take the constant out:

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

= 7 \cdot \frac{1}{3} \cdot \int u^{19} d u

Apply the Power Rule

: \frac{u ^{20}}{20}

= 7 \cdot \frac{1}{3} \cdot \frac{u ^{20}}{20}

Substitute back

u = x^{3} - 8

= 7 \cdot \frac{1}{3} \cdot \frac{( x ^{3} - 8 ) ^{20}}{20}

Simplify

7 \cdot \frac{1}{3} \cdot \frac{( x ^{3} - 8 ) ^{20}}{20} : \frac{7}{60} (x^{3} - 8)^{20}

= \frac{7}{60} (x^{3} - 8)^{20}

Add a constant to the solution

= \frac{7}{60} (x^{3} - 8)^{20} + C

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What is the integral of 7x^2(x^3-8)^{19} ?
The integral of 7x^2(x^3-8)^{19} is 7/60 (x^3-8)^{20}+C