What is the integral of (1+x^7)^57x^6 ?

The integral of (1+x^7)^57x^6 is x^7+(5x^{14})/2+(10x^{21})/3+(5x^{28})/2+x^{35}+(x^{42})/6+C

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integral of (1+x^7)^57x^6

\int (1 + x^{7})^{5} 7 x^{6} d x

x^{7} + \frac{5 x ^{14}}{2} + \frac{10 x ^{21}}{3} + \frac{5 x ^{28}}{2} + x^{35} + \frac{x ^{42}}{6} + C

Solution steps

\int (1 + x^{7})^{5} \cdot 7 x^{6} d x

Take the constant out:

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

= 7 \cdot \int (1 + x^{7})^{5} x^{6} d x

Expand

(1 + x^{7})^{5} x^{6} : x^{6} + 5 x^{13} + 10 x^{20} + 10 x^{27} + 5 x^{34} + x^{41}

= 7 \cdot \int x^{6} + 5 x^{13} + 10 x^{20} + 10 x^{27} + 5 x^{34} + x^{41} 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

= 7 (\int x^{6} d x + \int 5 x^{13} d x + \int 10 x^{20} d x + \int 10 x^{27} d x + \int 5 x^{34} d x + \int x^{41} d x)

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

\int 5 x^{13} d x = \frac{5 x ^{14}}{14}

\int 10 x^{20} d x = \frac{10 x ^{21}}{21}

\int 10 x^{27} d x = \frac{5 x ^{28}}{14}

\int 5 x^{34} d x = \frac{x ^{35}}{7}

\int x^{41} d x = \frac{x ^{42}}{42}

= 7 (\frac{x ^{7}}{7} + \frac{5 x ^{14}}{14} + \frac{10 x ^{21}}{21} + \frac{5 x ^{28}}{14} + \frac{x ^{35}}{7} + \frac{x ^{42}}{42})

Simplify

7 (\frac{x ^{7}}{7} + \frac{5 x ^{14}}{14} + \frac{10 x ^{21}}{21} + \frac{5 x ^{28}}{14} + \frac{x ^{35}}{7} + \frac{x ^{42}}{42}) : x^{7} + \frac{5 x ^{14}}{2} + \frac{10 x ^{21}}{3} + \frac{5 x ^{28}}{2} + x^{35} + \frac{x ^{42}}{6}

= x^{7} + \frac{5 x ^{14}}{2} + \frac{10 x ^{21}}{3} + \frac{5 x ^{28}}{2} + x^{35} + \frac{x ^{42}}{6}

Add a constant to the solution

= x^{7} + \frac{5 x ^{14}}{2} + \frac{10 x ^{21}}{3} + \frac{5 x ^{28}}{2} + x^{35} + \frac{x ^{42}}{6} + C

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What is the integral of (1+x^7)^57x^6 ?
The integral of (1+x^7)^57x^6 is x^7+(5x^{14})/2+(10x^{21})/3+(5x^{28})/2+x^{35}+(x^{42})/6+C