What is the integral of (1+x^5)^35x^4 ?

The integral of (1+x^5)^35x^4 is 1/4 (1+x^5)^4+C

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integral of (1+x^5)^35x^4

\int (1 + x^{5})^{3} 5 x^{4} d x

\frac{1}{4} (1 + x^{5})^{4} + C

Solution steps

\int (1 + x^{5})^{3} \cdot 5 x^{4} d x

Take the constant out:

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

= 5 \cdot \int (1 + x^{5})^{3} x^{4} d x

Apply u-substitution

: \int \frac{u ^{3}}{5} d u

= 5 \cdot \int \frac{u ^{3}}{5} d u

Take the constant out:

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

= 5 \cdot \frac{1}{5} \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= 5 \cdot \frac{1}{5} \cdot \frac{u ^{4}}{4}

Substitute back

u = 1 + x^{5}

= 5 \cdot \frac{1}{5} \cdot \frac{( 1 + x ^{5} ) ^{4}}{4}

Simplify

5 \cdot \frac{1}{5} \cdot \frac{( 1 + x ^{5} ) ^{4}}{4} : \frac{1}{4} (1 + x^{5})^{4}

= \frac{1}{4} (1 + x^{5})^{4}

Add a constant to the solution

= \frac{1}{4} (1 + x^{5})^{4} + C

\int \frac{x ^{5}}{x ^{2} + 5} d x

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What is the integral of (1+x^5)^35x^4 ?
The integral of (1+x^5)^35x^4 is 1/4 (1+x^5)^4+C