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

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

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

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

\frac{x ^{48}}{16} + x^{47} + \frac{15 x ^{46}}{2} + 35 x^{45} + \frac{455 x ^{44}}{4} + 273 x^{43} + \frac{1001 x ^{42}}{2} + 715 x^{41} + \frac{6435 x ^{40}}{8} + 715 x^{39} + \frac{1001 x ^{38}}{2} + 273 x^{37} + \frac{455 x ^{36}}{4} + 35 x^{35} + \frac{15 x ^{34}}{2} + x^{33} + \frac{x ^{32}}{16} + C

Solution steps

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

Expand

(x^{3} + x^{2})^{15} (3 x^{2} + 2 x) : 3 x^{47} + 47 x^{46} + 345 x^{45} + 1575 x^{44} + 5005 x^{43} + 11739 x^{42} + 21021 x^{41} + 29315 x^{40} + 32175 x^{39} + 27885 x^{38} + 19019 x^{37} + 10101 x^{36} + 4095 x^{35} + 1225 x^{34} + 255 x^{33} + 33 x^{32} + 2 x^{31}

= \int 3 x^{47} + 47 x^{46} + 345 x^{45} + 1575 x^{44} + 5005 x^{43} + 11739 x^{42} + 21021 x^{41} + 29315 x^{40} + 32175 x^{39} + 27885 x^{38} + 19019 x^{37} + 10101 x^{36} + 4095 x^{35} + 1225 x^{34} + 255 x^{33} + 33 x^{32} + 2 x^{31} 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

= \int 3 x^{47} d x + \int 47 x^{46} d x + \int 345 x^{45} d x + \int 1575 x^{44} d x + \int 5005 x^{43} d x + \int 11739 x^{42} d x + \int 21021 x^{41} d x + \int 29315 x^{40} d x + \int 32175 x^{39} d x + \int 27885 x^{38} d x + \int 19019 x^{37} d x + \int 10101 x^{36} d x + \int 4095 x^{35} d x + \int 1225 x^{34} d x + \int 255 x^{33} d x + \int 33 x^{32} d x + \int 2 x^{31} d x

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

\int 47 x^{46} d x = x^{47}

\int 345 x^{45} d x = \frac{15 x ^{46}}{2}

\int 1575 x^{44} d x = 35 x^{45}

\int 5005 x^{43} d x = \frac{455 x ^{44}}{4}

\int 11739 x^{42} d x = 273 x^{43}

\int 21021 x^{41} d x = \frac{1001 x ^{42}}{2}

\int 29315 x^{40} d x = 715 x^{41}

\int 32175 x^{39} d x = \frac{6435 x ^{40}}{8}

\int 27885 x^{38} d x = 715 x^{39}

\int 19019 x^{37} d x = \frac{1001 x ^{38}}{2}

\int 10101 x^{36} d x = 273 x^{37}

\int 4095 x^{35} d x = \frac{455 x ^{36}}{4}

\int 1225 x^{34} d x = 35 x^{35}

\int 255 x^{33} d x = \frac{15 x ^{34}}{2}

\int 33 x^{32} d x = x^{33}

\int 2 x^{31} d x = \frac{x ^{32}}{16}

= \frac{x ^{48}}{16} + x^{47} + \frac{15 x ^{46}}{2} + 35 x^{45} + \frac{455 x ^{44}}{4} + 273 x^{43} + \frac{1001 x ^{42}}{2} + 715 x^{41} + \frac{6435 x ^{40}}{8} + 715 x^{39} + \frac{1001 x ^{38}}{2} + 273 x^{37} + \frac{455 x ^{36}}{4} + 35 x^{35} + \frac{15 x ^{34}}{2} + x^{33} + \frac{x ^{32}}{16}

Add a constant to the solution

= \frac{x ^{48}}{16} + x^{47} + \frac{15 x ^{46}}{2} + 35 x^{45} + \frac{455 x ^{44}}{4} + 273 x^{43} + \frac{1001 x ^{42}}{2} + 715 x^{41} + \frac{6435 x ^{40}}{8} + 715 x^{39} + \frac{1001 x ^{38}}{2} + 273 x^{37} + \frac{455 x ^{36}}{4} + 35 x^{35} + \frac{15 x ^{34}}{2} + x^{33} + \frac{x ^{32}}{16} + C

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