What is the integral of (s^2+1)(s^3+3s)^4 ?

The integral of (s^2+1)(s^3+3s)^4 is 1/15 (s^3+3s)^5+C

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integral of (s^2+1)(s^3+3s)^4

\int (s^{2} + 1) (s^{3} + 3 s)^{4} d s

\frac{1}{15} (s^{3} + 3 s)^{5} + C

Solution steps

\int (s^{2} + 1) (s^{3} + 3 s)^{4} d s

Apply u-substitution

: \int \frac{u ^{4}}{3} d u

= \int \frac{u ^{4}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int u^{4} d u

Apply the Power Rule

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

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

Substitute back

u = s^{3} + 3 s

= \frac{1}{3} \cdot \frac{( s ^{3} + 3 s ) ^{5}}{5}

Simplify

\frac{1}{3} \cdot \frac{( s ^{3} + 3 s ) ^{5}}{5} : \frac{1}{15} (s^{3} + 3 s)^{5}

= \frac{1}{15} (s^{3} + 3 s)^{5}

Add a constant to the solution

= \frac{1}{15} (s^{3} + 3 s)^{5} + C

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What is the integral of (s^2+1)(s^3+3s)^4 ?
The integral of (s^2+1)(s^3+3s)^4 is 1/15 (s^3+3s)^5+C