What is the integral of 30(1+x^2)(7+15x+5x^3)^{10} ?

The integral of 30(1+x^2)(7+15x+5x^3)^{10} is 2/11 (5x^3+15x+7)^{11}+C

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integral of 30(1+x^2)(7+15x+5x^3)^{10}

\int 30 (1 + x^{2}) (7 + 15 x + 5 x^{3})^{10} d x

\frac{2}{11} (5 x^{3} + 15 x + 7)^{11} + C

Solution steps

\int 30 (1 + x^{2}) (7 + 15 x + 5 x^{3})^{10} d x

Take the constant out:

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

= 30 \cdot \int (1 + x^{2}) (7 + 15 x + 5 x^{3})^{10} d x

Apply u-substitution

= 30 \cdot \int \frac{u ^{10}}{15} d u

Take the constant out:

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

= 30 \cdot \frac{1}{15} \cdot \int u^{10} d u

Apply the Power Rule

= 30 \cdot \frac{1}{15} \cdot \frac{u ^{11}}{11}

Substitute back

u = 7 + 15 x + 5 x^{3}

= 30 \cdot \frac{1}{15} \cdot \frac{( 7 + 15 x + 5 x ^{3} ) ^{11}}{11}

Simplify

30 \cdot \frac{1}{15} \cdot \frac{( 7 + 15 x + 5 x ^{3} ) ^{11}}{11} : \frac{2}{11} (5 x^{3} + 15 x + 7)^{11}

= \frac{2}{11} (5 x^{3} + 15 x + 7)^{11}

Add a constant to the solution

= \frac{2}{11} (5 x^{3} + 15 x + 7)^{11} + C

What is the integral of 30(1+x^2)(7+15x+5x^3)^{10} ?
The integral of 30(1+x^2)(7+15x+5x^3)^{10} is 2/11 (5x^3+15x+7)^{11}+C