What is the integral of (3+7x^2)^9*5x ?

The integral of (3+7x^2)^9*5x is 1/28 (3+7x^2)^{10}+C

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integral of (3+7x^2)^9*5x

\int (3 + 7 x^{2})^{9} \cdot 5 x d x

\frac{1}{28} (3 + 7 x^{2})^{10} + C

Solution steps

\int (3 + 7 x^{2})^{9} \cdot 5 x d x

Take the constant out:

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

= 5 \cdot \int (3 + 7 x^{2})^{9} x d x

Apply u-substitution

: \int \frac{u ^{9}}{14} d u

= 5 \cdot \int \frac{u ^{9}}{14} d u

Take the constant out:

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

= 5 \cdot \frac{1}{14} \cdot \int u^{9} d u

Apply the Power Rule

: \frac{u ^{10}}{10}

= 5 \cdot \frac{1}{14} \cdot \frac{u ^{10}}{10}

Substitute back

u = 3 + 7 x^{2}

= 5 \cdot \frac{1}{14} \cdot \frac{( 3 + 7 x ^{2} ) ^{10}}{10}

Simplify

5 \cdot \frac{1}{14} \cdot \frac{( 3 + 7 x ^{2} ) ^{10}}{10} : \frac{1}{28} (3 + 7 x^{2})^{10}

= \frac{1}{28} (3 + 7 x^{2})^{10}

Add a constant to the solution

= \frac{1}{28} (3 + 7 x^{2})^{10} + C

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What is the integral of (3+7x^2)^9*5x ?
The integral of (3+7x^2)^9*5x is 1/28 (3+7x^2)^{10}+C