What is the integral of 5arctan(1/x) ?

The integral of 5arctan(1/x) is 5(xarctan(1/x)+1/2 ln|1+x^2|)+C

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integral of 5arctan(1/x)

\int 5 arctan (\frac{1}{x}) d x

5 (x arctan (\frac{1}{x}) + \frac{1}{2} ln 1 + x^{2}) + C

Solution steps

\int 5 arctan (\frac{1}{x}) d x

Take the constant out:

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

= 5 \cdot \int arctan (\frac{1}{x}) d x

Apply Integration By Parts

: x arctan (\frac{1}{x}) - \int - \frac{x}{1 + x ^{2}} d x

= 5 (x arctan (\frac{1}{x}) - \int - \frac{x}{1 + x ^{2}} d x)

\int - \frac{x}{1 + x ^{2}} d x = - \frac{1}{2} ln 1 + x^{2}

= 5 (x arctan (\frac{1}{x}) - (- \frac{1}{2} ln 1 + x^{2}))

Simplify

= 5 (x arctan (\frac{1}{x}) + \frac{1}{2} ln 1 + x^{2})

Add a constant to the solution

= 5 (x arctan (\frac{1}{x}) + \frac{1}{2} ln 1 + x^{2}) + C

\int x^{2} ln ∣ x ∣ d x

x \to 1 lim (x (x - 1) ln (x + 1))

\int (2 x^{2})^{2} d x

\frac{1}{2} e^{t} - \frac{1}{2} cos (t) - \frac{1}{2} sin (t)

2 e^{2 t}

What is the integral of 5arctan(1/x) ?
The integral of 5arctan(1/x) is 5(xarctan(1/x)+1/2 ln|1+x^2|)+C