What is the integral of (2+x^6)^86x^5 ?

The integral of (2+x^6)^86x^5 is

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integral of (2+x^6)^86x^5

\int (2 + x^{6})^{8} 6 x^{5} d x

6 (\frac{128 x ^{6}}{3} + \frac{256 x ^{12}}{3} + \frac{896 x ^{18}}{9} + \frac{224 x ^{24}}{3} + \frac{112 x ^{30}}{3} + \frac{112 x ^{36}}{9} + \frac{8 x ^{42}}{3} + \frac{x ^{48}}{3} + \frac{x ^{54}}{54}) + C

Solution steps

\int (2 + x^{6})^{8} \cdot 6 x^{5} d x

Take the constant out:

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

= 6 \cdot \int (2 + x^{6})^{8} x^{5} d x

Expand

(2 + x^{6})^{8} x^{5} : 256 x^{5} + 1024 x^{11} + 1792 x^{17} + 1792 x^{23} + 1120 x^{29} + 448 x^{35} + 112 x^{41} + 16 x^{47} + x^{53}

= 6 \cdot \int 256 x^{5} + 1024 x^{11} + 1792 x^{17} + 1792 x^{23} + 1120 x^{29} + 448 x^{35} + 112 x^{41} + 16 x^{47} + x^{53} 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

= 6 (\int 256 x^{5} d x + \int 1024 x^{11} d x + \int 1792 x^{17} d x + \int 1792 x^{23} d x + \int 1120 x^{29} d x + \int 448 x^{35} d x + \int 112 x^{41} d x + \int 16 x^{47} d x + \int x^{53} d x)

\int 256 x^{5} d x = \frac{128 x ^{6}}{3}

\int 1024 x^{11} d x = \frac{256 x ^{12}}{3}

\int 1792 x^{17} d x = \frac{896 x ^{18}}{9}

\int 1792 x^{23} d x = \frac{224 x ^{24}}{3}

\int 1120 x^{29} d x = \frac{112 x ^{30}}{3}

\int 448 x^{35} d x = \frac{112 x ^{36}}{9}

\int 112 x^{41} d x = \frac{8 x ^{42}}{3}

\int 16 x^{47} d x = \frac{x ^{48}}{3}

\int x^{53} d x = \frac{x ^{54}}{54}

= 6 (\frac{128 x ^{6}}{3} + \frac{256 x ^{12}}{3} + \frac{896 x ^{18}}{9} + \frac{224 x ^{24}}{3} + \frac{112 x ^{30}}{3} + \frac{112 x ^{36}}{9} + \frac{8 x ^{42}}{3} + \frac{x ^{48}}{3} + \frac{x ^{54}}{54})

Add a constant to the solution

= 6 (\frac{128 x ^{6}}{3} + \frac{256 x ^{12}}{3} + \frac{896 x ^{18}}{9} + \frac{224 x ^{24}}{3} + \frac{112 x ^{30}}{3} + \frac{112 x ^{36}}{9} + \frac{8 x ^{42}}{3} + \frac{x ^{48}}{3} + \frac{x ^{54}}{54}) + C

\int (\frac{x ^{3} - 6 x + 5}{x}) d x

\frac{d}{d x} (tan (I n x))

\frac{d}{d x} (\frac{2 x + 1}{3})

\int x^{- 8} (x + 1) d x

\frac{sin ( 8 x )}{13 x}