What is the integral of 2/x+5csc^2(x)+x ?

The integral of 2/x+5csc^2(x)+x is 2ln|x|-5cot(x)+(x^2)/2+C

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integral of 2/x+5csc^2(x)+x

\int \frac{2}{x} + 5 csc^{2} (x) + x d x

2 ln ∣ x ∣ - 5 cot (x) + \frac{x ^{2}}{2} + C

Solution steps

\int \frac{2}{x} + 5 csc^{2} (x) + 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 \frac{2}{x} d x + \int 5 csc^{2} (x) d x + \int x d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int 5 csc^{2} (x) d x = - 5 cot (x)

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

= 2 ln ∣ x ∣ - 5 cot (x) + \frac{x ^{2}}{2}

Add a constant to the solution

= 2 ln ∣ x ∣ - 5 cot (x) + \frac{x ^{2}}{2} + C

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What is the integral of 2/x+5csc^2(x)+x ?
The integral of 2/x+5csc^2(x)+x is 2ln|x|-5cot(x)+(x^2)/2+C