What is the integral of u^{-5}(5-u^5+u^{10}) ?

The integral of u^{-5}(5-u^5+u^{10}) is -5/(4u^4)-u+(u^6)/6+C

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integral of u^{-5}(5-u^5+u^{10})

\int u^{- 5} (5 - u^{5} + u^{10}) d u

- \frac{5}{4 u ^{4}} - u + \frac{u ^{6}}{6} + C

Solution steps

\int u^{- 5} (5 - u^{5} + u^{10}) d u

Expand

u^{- 5} (5 - u^{5} + u^{10}) : \frac{5}{u ^{5}} - 1 + u^{5}

= \int \frac{5}{u ^{5}} - 1 + u^{5} d u

Apply the Sum Rule:

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

= \int \frac{5}{u ^{5}} d u - \int 1 d u + \int u^{5} d u

\int \frac{5}{u ^{5}} d u = - \frac{5}{4 u ^{4}}

\int 1 d u = u

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

= - \frac{5}{4 u ^{4}} - u + \frac{u ^{6}}{6}

Add a constant to the solution

= - \frac{5}{4 u ^{4}} - u + \frac{u ^{6}}{6} + C

\int x^{18} d x

y^{^{''}} + 4 y^{^{'}} + 13 y = 0, y (0) = 3, y^{^{'}} (0) = - 5

\int_{0}^{2 π} 5 d x

x \to 0 + lim (- x ln (x))

\int \frac{x - 2}{x ^{2} - 4 x + 3} d x

What is the integral of u^{-5}(5-u^5+u^{10}) ?
The integral of u^{-5}(5-u^5+u^{10}) is -5/(4u^4)-u+(u^6)/6+C