What is the integral of 5x^3(x^4+5)^4 ?

The integral of 5x^3(x^4+5)^4 is 1/4 (x^4+5)^5+C

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integral of 5x^3(x^4+5)^4

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 5 \cdot \int \frac{u ^{4}}{4} d u

Take the constant out:

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

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

Apply the Power Rule

: \frac{u ^{5}}{5}

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

Substitute back

u = x^{4} + 5

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 5x^3(x^4+5)^4 ?
The integral of 5x^3(x^4+5)^4 is 1/4 (x^4+5)^5+C