What is the integral of (x^3(2+x^4)^5) ?

The integral of (x^3(2+x^4)^5) is 1/24 (2+x^4)^6+C

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integral of (x^3(2+x^4)^5)

\int (x^{3} (2 + x^{4})^{5}) d x

\frac{1}{24} (2 + x^{4})^{6} + C

Solution steps

\int x^{3} (2 + x^{4})^{5} d x

Apply u-substitution

: \int \frac{u ^{5}}{4} d u

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

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{1}{4} \cdot \frac{u ^{6}}{6}

Substitute back

u = 2 + x^{4}

= \frac{1}{4} \cdot \frac{( 2 + x ^{4} ) ^{6}}{6}

Simplify

\frac{1}{4} \cdot \frac{( 2 + x ^{4} ) ^{6}}{6} : \frac{1}{24} (2 + x^{4})^{6}

= \frac{1}{24} (2 + x^{4})^{6}

Add a constant to the solution

= \frac{1}{24} (2 + x^{4})^{6} + C

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What is the integral of (x^3(2+x^4)^5) ?
The integral of (x^3(2+x^4)^5) is 1/24 (2+x^4)^6+C