What is the integral of x^3(2-x^2)^{12} ?

The integral of x^3(2-x^2)^{12} is -1/364 (-13(2-x^2)^{14}+28(2-x^2)^{13})+C

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integral of x^3(2-x^2)^{12}

\int x^{3} (2 - x^{2})^{12} d x

- \frac{1}{364} (- 13 (2 - x^{2})^{14} + 28 (2 - x^{2})^{13}) + C

Solution steps

\int x^{3} (2 - x^{2})^{12} d x

Apply u-substitution

= \int - \frac{( - u + 2 ) u ^{12}}{2} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int (- u + 2) u^{12} d u

Expand

(- u + 2) u^{12} : - u^{13} + 2 u^{12}

= - \frac{1}{2} \cdot \int - u^{13} + 2 u^{12} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \frac{1}{2} (- \int u^{13} d u + \int 2 u^{12} d u)

\int u^{13} d u = \frac{u ^{14}}{14}

\int 2 u^{12} d u = \frac{2 u ^{13}}{13}

= - \frac{1}{2} (- \frac{u ^{14}}{14} + \frac{2 u ^{13}}{13})

Substitute back

u = 2 - x^{2}

= - \frac{1}{2} (- \frac{( 2 - x ^{2} ) ^{14}}{14} + \frac{2 ( 2 - x ^{2} ) ^{13}}{13})

Simplify

- \frac{1}{2} (- \frac{( 2 - x ^{2} ) ^{14}}{14} + \frac{2 ( 2 - x ^{2} ) ^{13}}{13}) : - \frac{1}{364} (- 13 (2 - x^{2})^{14} + 28 (2 - x^{2})^{13})

= - \frac{1}{364} (- 13 (2 - x^{2})^{14} + 28 (2 - x^{2})^{13})

Add a constant to the solution

= - \frac{1}{364} (- 13 (2 - x^{2})^{14} + 28 (2 - x^{2})^{13}) + C

What is the integral of x^3(2-x^2)^{12} ?
The integral of x^3(2-x^2)^{12} is -1/364 (-13(2-x^2)^{14}+28(2-x^2)^{13})+C