What is the integral of x(x+1)^5 ?

The integral of x(x+1)^5 is (x^7)/7+(5x^6)/6+2x^5+(5x^4)/2+(5x^3)/3+(x^2)/2+C

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integral of x(x+1)^5

\int x (x + 1)^{5} d x

\frac{x ^{7}}{7} + \frac{5 x ^{6}}{6} + 2 x^{5} + \frac{5 x ^{4}}{2} + \frac{5 x ^{3}}{3} + \frac{x ^{2}}{2} + C

Solution steps

\int x (x + 1)^{5} d x

Expand

x (x + 1)^{5} : x^{6} + 5 x^{5} + 10 x^{4} + 10 x^{3} + 5 x^{2} + x

= \int x^{6} + 5 x^{5} + 10 x^{4} + 10 x^{3} + 5 x^{2} + 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 5 x^{5} d x + \int 10 x^{4} d x + \int 10 x^{3} d x + \int 5 x^{2} d x + \int x d x

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

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

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

\int 10 x^{3} d x = \frac{5 x ^{4}}{2}

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

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

= \frac{x ^{7}}{7} + \frac{5 x ^{6}}{6} + 2 x^{5} + \frac{5 x ^{4}}{2} + \frac{5 x ^{3}}{3} + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{7}}{7} + \frac{5 x ^{6}}{6} + 2 x^{5} + \frac{5 x ^{4}}{2} + \frac{5 x ^{3}}{3} + \frac{x ^{2}}{2} + C

\int (3 x - 4) e^{(- x)} d x

\frac{d}{d x} (2 x^{4} + x^{2} - x^{2} + 4)

\frac{\partial}{\partial x} (y \cdot e^{x y})

\frac{s + 1}{s ^{2} + 2}

x^{2} - 9 x

What is the integral of x(x+1)^5 ?
The integral of x(x+1)^5 is (x^7)/7+(5x^6)/6+2x^5+(5x^4)/2+(5x^3)/3+(x^2)/2+C