What is the integral of x(x+5)^{21} ?

The integral of x(x+5)^{21} is 1/506 (x+5)^{22}(22x-5)+C

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integral of x(x+5)^{21}

\int x (x + 5)^{21} d x

\frac{1}{506} (x + 5)^{22} (22 x - 5) + C

Solution steps

\int x (x + 5)^{21} d x

Apply u-substitution

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Expand

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

\int u^{\frac{2}{21}} d u = \frac{21}{23} u^{\frac{23}{21}}

= \frac{1}{21} (\frac{21}{23} u^{\frac{23}{21}} - \frac{105}{22} u^{\frac{22}{21}})

Substitute back

u = (x + 5)^{21}

= \frac{1}{21} (\frac{21}{23} ((x + 5)^{21})^{\frac{23}{21}} - \frac{105}{22} ((x + 5)^{21})^{\frac{22}{21}})

Simplify

\frac{1}{21} (\frac{21}{23} ((x + 5)^{21})^{\frac{23}{21}} - \frac{105}{22} ((x + 5)^{21})^{\frac{22}{21}}) : \frac{1}{506} (x + 5)^{22} (22 x - 5)

= \frac{1}{506} (x + 5)^{22} (22 x - 5)

Add a constant to the solution

= \frac{1}{506} (x + 5)^{22} (22 x - 5) + C

What is the integral of x(x+5)^{21} ?
The integral of x(x+5)^{21} is 1/506 (x+5)^{22}(22x-5)+C