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Popular >

integral of x^4(3kx^5+2)^7

Solution

\int x^{4} (3 k x^{5} + 2)^{7} d x

Solution

\frac{2187 k ^{7} x ^{40}}{40} + \frac{1458 k ^{6} x ^{35}}{5} + \frac{3402 k ^{5} x ^{30}}{5} + \frac{4536 k ^{4} x ^{25}}{5} + 756 k^{3} x^{20} + \frac{2016 k ^{2} x ^{15}}{5} + \frac{672 k x ^{10}}{5} + \frac{128 x ^{5}}{5} + C

Solution steps

\int x^{4} (3 k x^{5} + 2)^{7} d x

Expand

x^{4} (3 k x^{5} + 2)^{7} : 2187 k^{7} x^{39} + 10206 k^{6} x^{34} + 20412 k^{5} x^{29} + 22680 k^{4} x^{24} + 15120 k^{3} x^{19} + 6048 k^{2} x^{14} + 1344 k x^{9} + 128 x^{4}

= \int 2187 k^{7} x^{39} + 10206 k^{6} x^{34} + 20412 k^{5} x^{29} + 22680 k^{4} x^{24} + 15120 k^{3} x^{19} + 6048 k^{2} x^{14} + 1344 k x^{9} + 128 x^{4} 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 2187 k^{7} x^{39} d x + \int 10206 k^{6} x^{34} d x + \int 20412 k^{5} x^{29} d x + \int 22680 k^{4} x^{24} d x + \int 15120 k^{3} x^{19} d x + \int 6048 k^{2} x^{14} d x + \int 1344 k x^{9} d x + \int 128 x^{4} d x

\int 2187 k^{7} x^{39} d x = \frac{2187 k ^{7} x ^{40}}{40}

\int 10206 k^{6} x^{34} d x = \frac{1458 k ^{6} x ^{35}}{5}

\int 20412 k^{5} x^{29} d x = \frac{3402 k ^{5} x ^{30}}{5}

\int 22680 k^{4} x^{24} d x = \frac{4536 k ^{4} x ^{25}}{5}

\int 15120 k^{3} x^{19} d x = 756 k^{3} x^{20}

\int 6048 k^{2} x^{14} d x = \frac{2016 k ^{2} x ^{15}}{5}

\int 1344 k x^{9} d x = \frac{672 k x ^{10}}{5}

\int 128 x^{4} d x = \frac{128 x ^{5}}{5}

= \frac{2187 k ^{7} x ^{40}}{40} + \frac{1458 k ^{6} x ^{35}}{5} + \frac{3402 k ^{5} x ^{30}}{5} + \frac{4536 k ^{4} x ^{25}}{5} + 756 k^{3} x^{20} + \frac{2016 k ^{2} x ^{15}}{5} + \frac{672 k x ^{10}}{5} + \frac{128 x ^{5}}{5}

Add a constant to the solution

= \frac{2187 k ^{7} x ^{40}}{40} + \frac{1458 k ^{6} x ^{35}}{5} + \frac{3402 k ^{5} x ^{30}}{5} + \frac{4536 k ^{4} x ^{25}}{5} + 756 k^{3} x^{20} + \frac{2016 k ^{2} x ^{15}}{5} + \frac{672 k x ^{10}}{5} + \frac{128 x ^{5}}{5} + C

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Frequently Asked Questions (FAQ)

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The integral of x^4(3kx^5+2)^7 is