What is the integral of 2x(x^2+19)^{14} ?

The integral of 2x(x^2+19)^{14} is 1/15 (x^2+19)^{15}+C

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integral of 2x(x^2+19)^{14}

\int 2 x (x^{2} + 19)^{14} d x

\frac{1}{15} (x^{2} + 19)^{15} + C

Solution steps

\int 2 x (x^{2} + 19)^{14} d x

Take the constant out:

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

= 2 \cdot \int x (x^{2} + 19)^{14} d x

Apply u-substitution

: \int \frac{u ^{14}}{2} d u

= 2 \cdot \int \frac{u ^{14}}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int u^{14} d u

Apply the Power Rule

: \frac{u ^{15}}{15}

= 2 \cdot \frac{1}{2} \cdot \frac{u ^{15}}{15}

Substitute back

u = x^{2} + 19

= 2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 19 ) ^{15}}{15}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{( x ^{2} + 19 ) ^{15}}{15} : \frac{1}{15} (x^{2} + 19)^{15}

= \frac{1}{15} (x^{2} + 19)^{15}

Add a constant to the solution

= \frac{1}{15} (x^{2} + 19)^{15} + C

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What is the integral of 2x(x^2+19)^{14} ?
The integral of 2x(x^2+19)^{14} is 1/15 (x^2+19)^{15}+C