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

The integral of (z^2+5)^3z is 1/8 (z^2+5)^4+C

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integral of (z^2+5)^3z

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

\frac{1}{8} (z^{2} + 5)^{4} + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{3}}{2} d u

= \int \frac{u ^{3}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u^{3} d u

Apply the Power Rule

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

= \frac{1}{2} \cdot \frac{u ^{4}}{4}

Substitute back

u = z^{2} + 5

= \frac{1}{2} \cdot \frac{( z ^{2} + 5 ) ^{4}}{4}

Simplify

\frac{1}{2} \cdot \frac{( z ^{2} + 5 ) ^{4}}{4} : \frac{1}{8} (z^{2} + 5)^{4}

= \frac{1}{8} (z^{2} + 5)^{4}

Add a constant to the solution

= \frac{1}{8} (z^{2} + 5)^{4} + C

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What is the integral of (z^2+5)^3z ?
The integral of (z^2+5)^3z is 1/8 (z^2+5)^4+C