What is the integral of x^3(1-5x^2)^{10} ?

The integral of x^3(1-5x^2)^{10} is 1/6600 (11(1-5x^2)^{12}-12(1-5x^2)^{11})+C

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integral of x^3(1-5x^2)^{10}

\int x^{3} (1 - 5 x^{2})^{10} d x

\frac{1}{6600} (11 (1 - 5 x^{2})^{12} - 12 (1 - 5 x^{2})^{11}) + C

Solution steps

\int x^{3} (1 - 5 x^{2})^{10} d x

Apply u-substitution

= \int \frac{u ^{10} ( u - 1 )}{50} d u

Take the constant out:

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

= \frac{1}{50} \cdot \int u^{10} (u - 1) d u

Expand

u^{10} (u - 1) : u^{11} - u^{10}

= \frac{1}{50} \cdot \int u^{11} - u^{10} 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}{50} (\int u^{11} d u - \int u^{10} d u)

\int u^{11} d u = \frac{u ^{12}}{12}

\int u^{10} d u = \frac{u ^{11}}{11}

= \frac{1}{50} (\frac{u ^{12}}{12} - \frac{u ^{11}}{11})

Substitute back

u = 1 - 5 x^{2}

= \frac{1}{50} (\frac{( 1 - 5 x ^{2} ) ^{12}}{12} - \frac{( 1 - 5 x ^{2} ) ^{11}}{11})

Simplify

\frac{1}{50} (\frac{( 1 - 5 x ^{2} ) ^{12}}{12} - \frac{( 1 - 5 x ^{2} ) ^{11}}{11}) : \frac{1}{6600} (11 (1 - 5 x^{2})^{12} - 12 (1 - 5 x^{2})^{11})

= \frac{1}{6600} (11 (1 - 5 x^{2})^{12} - 12 (1 - 5 x^{2})^{11})

Add a constant to the solution

= \frac{1}{6600} (11 (1 - 5 x^{2})^{12} - 12 (1 - 5 x^{2})^{11}) + C

What is the integral of x^3(1-5x^2)^{10} ?
The integral of x^3(1-5x^2)^{10} is 1/6600 (11(1-5x^2)^{12}-12(1-5x^2)^{11})+C