What is the integral of x(x+8)^5 ?

The integral of x(x+8)^5 is (x^7)/7+(20x^6)/3+128x^5+1280x^4+(20480x^3)/3+16384x^2+C

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integral of x(x+8)^5

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

\frac{x ^{7}}{7} + \frac{20 x ^{6}}{3} + 128 x^{5} + 1280 x^{4} + \frac{20480 x ^{3}}{3} + 16384 x^{2} + C

Solution steps

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

Expand

x (x + 8)^{5} : x^{6} + 40 x^{5} + 640 x^{4} + 5120 x^{3} + 20480 x^{2} + 32768 x

= \int x^{6} + 40 x^{5} + 640 x^{4} + 5120 x^{3} + 20480 x^{2} + 32768 x 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 40 x^{5} d x + \int 640 x^{4} d x + \int 5120 x^{3} d x + \int 20480 x^{2} d x + \int 32768 x d x

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

\int 40 x^{5} d x = \frac{20 x ^{6}}{3}

\int 640 x^{4} d x = 128 x^{5}

\int 5120 x^{3} d x = 1280 x^{4}

\int 20480 x^{2} d x = \frac{20480 x ^{3}}{3}

\int 32768 x d x = 16384 x^{2}

= \frac{x ^{7}}{7} + \frac{20 x ^{6}}{3} + 128 x^{5} + 1280 x^{4} + \frac{20480 x ^{3}}{3} + 16384 x^{2}

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{20 x ^{6}}{3} + 128 x^{5} + 1280 x^{4} + \frac{20480 x ^{3}}{3} + 16384 x^{2} + C

\frac{d}{d x} (- 3 cos^{2} (x))

y = x^{3} + 1

\frac{6 x + 8 x ^{2}}{( 3 + 8 x ) ^{2}}

\frac{d x}{d t} = - 2 x + t

x, at 4

What is the integral of x(x+8)^5 ?
The integral of x(x+8)^5 is (x^7)/7+(20x^6)/3+128x^5+1280x^4+(20480x^3)/3+16384x^2+C