What is the integral of (x^3(x^3+2))/(15) ?

The integral of (x^3(x^3+2))/(15) is 1/15 ((x^7)/7+(x^4)/2)+C

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integral of (x^3(x^3+2))/(15)

\int \frac{x ^{3} ( x ^{3} + 2 )}{15} d x

\frac{1}{15} (\frac{x ^{7}}{7} + \frac{x ^{4}}{2}) + C

Solution steps

\int \frac{x ^{3} ( x ^{3} + 2 )}{15} d x

Take the constant out:

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

= \frac{1}{15} \cdot \int x^{3} (x^{3} + 2) d x

Expand

x^{3} (x^{3} + 2) : x^{6} + 2 x^{3}

= \frac{1}{15} \cdot \int x^{6} + 2 x^{3} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{15} (\int x^{6} d x + \int 2 x^{3} d x)

\int x^{6} d x = \frac{x ^{7}}{7}

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

= \frac{1}{15} (\frac{x ^{7}}{7} + \frac{x ^{4}}{2})

Add a constant to the solution

= \frac{1}{15} (\frac{x ^{7}}{7} + \frac{x ^{4}}{2}) + C

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What is the integral of (x^3(x^3+2))/(15) ?
The integral of (x^3(x^3+2))/(15) is 1/15 ((x^7)/7+(x^4)/2)+C