What is the integral of 8x^3(x^4+7)^2 ?

The integral of 8x^3(x^4+7)^2 is 8((x^{12}}{12}+\frac{7x^8)/4+(49x^4)/4)+C

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integral of 8x^3(x^4+7)^2

\int 8 x^{3} (x^{4} + 7)^{2} d x

8 (\frac{x ^{12}}{12} + \frac{7 x ^{8}}{4} + \frac{49 x ^{4}}{4}) + C

Solution steps

\int 8 x^{3} (x^{4} + 7)^{2} d x

Take the constant out:

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

= 8 \cdot \int x^{3} (x^{4} + 7)^{2} d x

Expand

x^{3} (x^{4} + 7)^{2} : x^{11} + 14 x^{7} + 49 x^{3}

= 8 \cdot \int x^{11} + 14 x^{7} + 49 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

= 8 (\int x^{11} d x + \int 14 x^{7} d x + \int 49 x^{3} d x)

\int x^{11} d x = \frac{x ^{12}}{12}

\int 14 x^{7} d x = \frac{7 x ^{8}}{4}

\int 49 x^{3} d x = \frac{49 x ^{4}}{4}

= 8 (\frac{x ^{12}}{12} + \frac{7 x ^{8}}{4} + \frac{49 x ^{4}}{4})

Add a constant to the solution

= 8 (\frac{x ^{12}}{12} + \frac{7 x ^{8}}{4} + \frac{49 x ^{4}}{4}) + C

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What is the integral of 8x^3(x^4+7)^2 ?
The integral of 8x^3(x^4+7)^2 is 8((x^{12}}{12}+\frac{7x^8)/4+(49x^4)/4)+C