What is the integral of (x^{3/2}+3)^4sqrt(x) ?

The integral of (x^{3/2}+3)^4sqrt(x) is 2/15 (x^{3/2}+3)^5+C

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integral of (x^{3/2}+3)^4sqrt(x)

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

\frac{2}{15} (x^{\frac{3}{2}} + 3)^{5} + C

Solution steps

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

Apply u-substitution

: \int \frac{2 u ^{4}}{3} d u

= \int \frac{2 u ^{4}}{3} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int u^{4} d u

Apply the Power Rule

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

= \frac{2}{3} \cdot \frac{u ^{5}}{5}

Substitute back

u = x^{\frac{3}{2}} + 3

= \frac{2}{3} \cdot \frac{( x ^{\frac{3}{2}} + 3 ) ^{5}}{5}

Simplify

\frac{2}{3} \cdot \frac{( x ^{\frac{3}{2}} + 3 ) ^{5}}{5} : \frac{2}{15} (x^{\frac{3}{2}} + 3)^{5}

= \frac{2}{15} (x^{\frac{3}{2}} + 3)^{5}

Add a constant to the solution

= \frac{2}{15} (x^{\frac{3}{2}} + 3)^{5} + C

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What is the integral of (x^{3/2}+3)^4sqrt(x) ?
The integral of (x^{3/2}+3)^4sqrt(x) is 2/15 (x^{3/2}+3)^5+C