What is the integral of 5x(x^2-1)^{50} ?

The integral of 5x(x^2-1)^{50} is 5/102 (x^2-1)^{51}+C

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integral of 5x(x^2-1)^{50}

\int 5 x (x^{2} - 1)^{50} d x

\frac{5}{102} (x^{2} - 1)^{51} + C

Solution steps

\int 5 x (x^{2} - 1)^{50} d x

Take the constant out:

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

= 5 \cdot \int x (x^{2} - 1)^{50} d x

Apply u-substitution

: \int \frac{u ^{50}}{2} d u

= 5 \cdot \int \frac{u ^{50}}{2} d u

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot \int u^{50} d u

Apply the Power Rule

: \frac{u ^{51}}{51}

= 5 \cdot \frac{1}{2} \cdot \frac{u ^{51}}{51}

Substitute back

u = x^{2} - 1

= 5 \cdot \frac{1}{2} \cdot \frac{( x ^{2} - 1 ) ^{51}}{51}

Simplify

5 \cdot \frac{1}{2} \cdot \frac{( x ^{2} - 1 ) ^{51}}{51} : \frac{5}{102} (x^{2} - 1)^{51}

= \frac{5}{102} (x^{2} - 1)^{51}

Add a constant to the solution

= \frac{5}{102} (x^{2} - 1)^{51} + C

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What is the integral of 5x(x^2-1)^{50} ?
The integral of 5x(x^2-1)^{50} is 5/102 (x^2-1)^{51}+C