integral of 1/(sqrt(x))(sqrt(x)-pi)^5

Solution

\int \frac{1}{x} (x - π)^{5} d x

\frac{x ^{3}}{3} - 2 π x^{\frac{5}{2}} + 5 π^{2} x^{2} - \frac{20 π ^{3}}{3} x^{\frac{3}{2}} + 5 π^{4} x - 2 π^{5} x + C

Solution steps

\int \frac{1}{x} (x - π)^{5} d x

Expand

\frac{1}{x} (x - π)^{5} : x^{2} - 5 π x^{\frac{3}{2}} + 10 π^{2} x - 10 π^{3} x + 5 π^{4} - \frac{π ^{5}}{x}

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

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

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

\int 10 π^{2} x d x = 5 π^{2} x^{2}

\int 10 π^{3} x d x = \frac{20 π ^{3}}{3} x^{\frac{3}{2}}

\int 5 π^{4} d x = 5 π^{4} x

\int \frac{π ^{5}}{x} d x = 2 π^{5} x

= \frac{x ^{3}}{3} - 2 π x^{\frac{5}{2}} + 5 π^{2} x^{2} - \frac{20 π ^{3}}{3} x^{\frac{3}{2}} + 5 π^{4} x - 2 π^{5} x

Add a constant to the solution

= \frac{x ^{3}}{3} - 2 π x^{\frac{5}{2}} + 5 π^{2} x^{2} - \frac{20 π ^{3}}{3} x^{\frac{3}{2}} + 5 π^{4} x - 2 π^{5} x + C