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

The integral of x^3(5+x^4)^4 is 1/20 (5+x^4)^5+C

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

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

\frac{1}{20} (5 + x^{4})^{5} + C

Solution steps

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

Apply u-substitution

= \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

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

Apply the Power Rule

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

Substitute back

u = 5 + x^{4}

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

Simplify

\frac{1}{4} \cdot \frac{( 5 + x ^{4} ) ^{5}}{5} : \frac{1}{20} (5 + x^{4})^{5}

= \frac{1}{20} (5 + x^{4})^{5}

Add a constant to the solution

= \frac{1}{20} (5 + x^{4})^{5} + C

What is the integral of x^3(5+x^4)^4 ?
The integral of x^3(5+x^4)^4 is 1/20 (5+x^4)^5+C