What is the integral of (5x^2-5)(x^3-3x)^{20} ?

The integral of (5x^2-5)(x^3-3x)^{20} is 5/63 (x^3-3x)^{21}+C

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integral of (5x^2-5)(x^3-3x)^{20}

\int (5 x^{2} - 5) (x^{3} - 3 x)^{20} d x

\frac{5}{63} (x^{3} - 3 x)^{21} + C

Solution steps

\int (5 x^{2} - 5) (x^{3} - 3 x)^{20} d x

Apply u-substitution

= \int \frac{5 u ^{20}}{3} d u

Take the constant out:

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

= \frac{5}{3} \cdot \int u^{20} d u

Apply the Power Rule

= \frac{5}{3} \cdot \frac{u ^{21}}{21}

Substitute back

u = x^{3} - 3 x

= \frac{5}{3} \cdot \frac{( x ^{3} - 3 x ) ^{21}}{21}

Simplify

\frac{5}{3} \cdot \frac{( x ^{3} - 3 x ) ^{21}}{21} : \frac{5}{63} (x^{3} - 3 x)^{21}

= \frac{5}{63} (x^{3} - 3 x)^{21}

Add a constant to the solution

= \frac{5}{63} (x^{3} - 3 x)^{21} + C

What is the integral of (5x^2-5)(x^3-3x)^{20} ?
The integral of (5x^2-5)(x^3-3x)^{20} is 5/63 (x^3-3x)^{21}+C