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

The integral of x^2(x^3+2)^5 is 1/18 (x^3+2)^6+C

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

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

\frac{1}{18} (x^{3} + 2)^{6} + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{5}}{3} d u

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

Apply the Power Rule

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

= \frac{1}{3} \cdot \frac{u ^{6}}{6}

Substitute back

u = x^{3} + 2

= \frac{1}{3} \cdot \frac{( x ^{3} + 2 ) ^{6}}{6}

Simplify

\frac{1}{3} \cdot \frac{( x ^{3} + 2 ) ^{6}}{6} : \frac{1}{18} (x^{3} + 2)^{6}

= \frac{1}{18} (x^{3} + 2)^{6}

Add a constant to the solution

= \frac{1}{18} (x^{3} + 2)^{6} + C

\frac{d}{d x} (\frac{cot ^{2} ( 2 x )}{1 + x ^{2}})

2 x (x + 5)

\frac{d}{d t} (6 cos^{3} (t))

\int \frac{( 3 x ^{3} - 5 x ^{2} + 10 x - 3 )}{3 x + 1} d x

\frac{8}{( x ^{3} + x ^{2} + 1 ) ^{3}}

What is the integral of x^2(x^3+2)^5 ?
The integral of x^2(x^3+2)^5 is 1/18 (x^3+2)^6+C