What is the integral of (x+1+tan(sqrt(x)))/(sqrt(x)) ?

The integral of (x+1+tan(sqrt(x)))/(sqrt(x)) is 2/3 x^{3/2}+2sqrt(x)-2ln|cos(sqrt(x))|+C

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integral of (x+1+tan(sqrt(x)))/(sqrt(x))

\int \frac{x + 1 + tan ( x )}{x} d x

\frac{2}{3} x^{\frac{3}{2}} + 2 x - 2 ln cos (x) + C

Solution steps

\int \frac{x + 1 + tan ( x )}{x} d x

Expand

\frac{x + 1 + tan ( x )}{x} : x + \frac{1}{x} + \frac{tan ( x )}{x}

= \int x + \frac{1}{x} + \frac{tan ( 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 x d x + \int \frac{1}{x} d x + \int \frac{tan ( x )}{x} d x

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

\int \frac{1}{x} d x = 2 x

\int \frac{tan ( x )}{x} d x = - 2 ln cos (x)

= \frac{2}{3} x^{\frac{3}{2}} + 2 x - 2 ln cos (x)

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + 2 x - 2 ln cos (x) + C

What is the integral of (x+1+tan(sqrt(x)))/(sqrt(x)) ?
The integral of (x+1+tan(sqrt(x)))/(sqrt(x)) is 2/3 x^{3/2}+2sqrt(x)-2ln|cos(sqrt(x))|+C