What is the integral of x(x^5+2x^4) ?

The integral of x(x^5+2x^4) is (x^7)/7+(x^6)/3+C

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integral of x(x^5+2x^4)

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

\frac{x ^{7}}{7} + \frac{x ^{6}}{3} + C

Solution steps

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

Multiply by the conjugate of

x^{5} + 2 x^{4} : \frac{x ^{10} - 4 x ^{8}}{x ^{5} - 2 x ^{4}}

= \int x \frac{x ^{10} - 4 x ^{8}}{x ^{5} - 2 x ^{4}} d x

Simplify

x \frac{x ^{10} - 4 x ^{8}}{x ^{5} - 2 x ^{4}} : x^{5} (x + 2)

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

Expand

x^{5} (x + 2) : x^{6} + 2 x^{5}

= \int x^{6} + 2 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

= \int x^{6} d x + \int 2 x^{5} d x

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

\int 2 x^{5} d x = \frac{x ^{6}}{3}

= \frac{x ^{7}}{7} + \frac{x ^{6}}{3}

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{x ^{6}}{3} + C

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What is the integral of x(x^5+2x^4) ?
The integral of x(x^5+2x^4) is (x^7)/7+(x^6)/3+C