What is the integral of 3x^5(x^3+2)^{15} ?

The integral of 3x^5(x^3+2)^{15} is

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integral of 3x^5(x^3+2)^{15}

\int 3 x^{5} (x^{3} + 2)^{15} d x

\frac{x ^{51}}{17} + \frac{15 x ^{48}}{8} + 28 x^{45} + 260 x^{42} + 1680 x^{39} + 8008 x^{36} + 29120 x^{33} + 82368 x^{30} + 183040 x^{27} + 320320 x^{24} + 439296 x^{21} + 465920 x^{18} + 372736 x^{15} + 215040 x^{12} + 81920 x^{9} + 16384 x^{6} + C

Solution steps

\int 3 x^{5} (x^{3} + 2)^{15} d x

Take the constant out:

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

= 3 \cdot \int x^{5} (x^{3} + 2)^{15} d x

Expand

x^{5} (x^{3} + 2)^{15} : x^{50} + 30 x^{47} + 420 x^{44} + 3640 x^{41} + 21840 x^{38} + 96096 x^{35} + 320320 x^{32} + 823680 x^{29} + 1647360 x^{26} + 2562560 x^{23} + 3075072 x^{20} + 2795520 x^{17} + 1863680 x^{14} + 860160 x^{11} + 245760 x^{8} + 32768 x^{5}

= 3 \cdot \int x^{50} + 30 x^{47} + 420 x^{44} + 3640 x^{41} + 21840 x^{38} + 96096 x^{35} + 320320 x^{32} + 823680 x^{29} + 1647360 x^{26} + 2562560 x^{23} + 3075072 x^{20} + 2795520 x^{17} + 1863680 x^{14} + 860160 x^{11} + 245760 x^{8} + 32768 x^{5} 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

= 3 (\int x^{50} d x + \int 30 x^{47} d x + \int 420 x^{44} d x + \int 3640 x^{41} d x + \int 21840 x^{38} d x + \int 96096 x^{35} d x + \int 320320 x^{32} d x + \int 823680 x^{29} d x + \int 1647360 x^{26} d x + \int 2562560 x^{23} d x + \int 3075072 x^{20} d x + \int 2795520 x^{17} d x + \int 1863680 x^{14} d x + \int 860160 x^{11} d x + \int 245760 x^{8} d x + \int 32768 x^{5} d x)

\int x^{50} d x = \frac{x ^{51}}{51}

\int 30 x^{47} d x = \frac{5 x ^{48}}{8}

\int 420 x^{44} d x = \frac{28 x ^{45}}{3}

\int 3640 x^{41} d x = \frac{260 x ^{42}}{3}

\int 21840 x^{38} d x = 560 x^{39}

\int 96096 x^{35} d x = \frac{8008 x ^{36}}{3}

\int 320320 x^{32} d x = \frac{29120 x ^{33}}{3}

\int 823680 x^{29} d x = 27456 x^{30}

\int 1647360 x^{26} d x = \frac{183040 x ^{27}}{3}

\int 2562560 x^{23} d x = \frac{320320 x ^{24}}{3}

\int 3075072 x^{20} d x = 146432 x^{21}

\int 2795520 x^{17} d x = \frac{465920 x ^{18}}{3}

\int 1863680 x^{14} d x = \frac{372736 x ^{15}}{3}

\int 860160 x^{11} d x = 71680 x^{12}

\int 245760 x^{8} d x = \frac{81920 x ^{9}}{3}

\int 32768 x^{5} d x = \frac{16384 x ^{6}}{3}

= 3 (\frac{x ^{51}}{51} + \frac{5 x ^{48}}{8} + \frac{28 x ^{45}}{3} + \frac{260 x ^{42}}{3} + 560 x^{39} + \frac{8008 x ^{36}}{3} + \frac{29120 x ^{33}}{3} + 27456 x^{30} + \frac{183040 x ^{27}}{3} + \frac{320320 x ^{24}}{3} + 146432 x^{21} + \frac{465920 x ^{18}}{3} + \frac{372736 x ^{15}}{3} + 71680 x^{12} + \frac{81920 x ^{9}}{3} + \frac{16384 x ^{6}}{3})

Simplify

3 (\frac{x ^{51}}{51} + \frac{5 x ^{48}}{8} + \frac{28 x ^{45}}{3} + \frac{260 x ^{42}}{3} + 560 x^{39} + \frac{8008 x ^{36}}{3} + \frac{29120 x ^{33}}{3} + 27456 x^{30} + \frac{183040 x ^{27}}{3} + \frac{320320 x ^{24}}{3} + 146432 x^{21} + \frac{465920 x ^{18}}{3} + \frac{372736 x ^{15}}{3} + 71680 x^{12} + \frac{81920 x ^{9}}{3} + \frac{16384 x ^{6}}{3}) : \frac{x ^{51}}{17} + \frac{15 x ^{48}}{8} + 28 x^{45} + 260 x^{42} + 1680 x^{39} + 8008 x^{36} + 29120 x^{33} + 82368 x^{30} + 183040 x^{27} + 320320 x^{24} + 439296 x^{21} + 465920 x^{18} + 372736 x^{15} + 215040 x^{12} + 81920 x^{9} + 16384 x^{6}

= \frac{x ^{51}}{17} + \frac{15 x ^{48}}{8} + 28 x^{45} + 260 x^{42} + 1680 x^{39} + 8008 x^{36} + 29120 x^{33} + 82368 x^{30} + 183040 x^{27} + 320320 x^{24} + 439296 x^{21} + 465920 x^{18} + 372736 x^{15} + 215040 x^{12} + 81920 x^{9} + 16384 x^{6}

Add a constant to the solution

= \frac{x ^{51}}{17} + \frac{15 x ^{48}}{8} + 28 x^{45} + 260 x^{42} + 1680 x^{39} + 8008 x^{36} + 29120 x^{33} + 82368 x^{30} + 183040 x^{27} + 320320 x^{24} + 439296 x^{21} + 465920 x^{18} + 372736 x^{15} + 215040 x^{12} + 81920 x^{9} + 16384 x^{6} + C

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