What is the integral of 1/x*cos^2(x) ?

The integral of 1/x*cos^2(x) is 1/2 (ln|x|+\Ci(2x))+C

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integral of 1/x*cos^2(x)

integral

\frac{1}{x} \cdot cos^{2} (x)

\frac{1}{2} (ln ∣ x ∣ + Ci (2 x)) + C

Solution steps

\int \frac{1}{x} cos^{2} (x) d x

Simplify

\frac{1}{x} cos^{2} (x) : \frac{cos ^{2} ( x )}{x}

= \int \frac{cos ^{2} ( x )}{x} d x

Rewrite using trig identities

= \int \frac{1 + cos ( 2 x )}{2 x} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1 + cos ( 2 x )}{x} d x

Expand

\frac{1 + cos ( 2 x )}{x} : \frac{1}{x} + \frac{cos ( 2 x )}{x}

= \frac{1}{2} \cdot \int \frac{1}{x} + \frac{cos ( 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

= \frac{1}{2} (\int \frac{1}{x} d x + \int \frac{cos ( 2 x )}{x} d x)

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{cos ( 2 x )}{x} d x = Ci (2 x)

= \frac{1}{2} (ln ∣ x ∣ + Ci (2 x))

Add a constant to the solution

= \frac{1}{2} (ln ∣ x ∣ + Ci (2 x)) + C

1 + e^{- x} sin (4 x)

2 x^{- 3}

(4 - x^{2})^{1.5}

(1 - x^{2})^{3}

\frac{( x ^{2} - 1 )}{( x ^{2} + 1 )}

What is the integral of 1/x*cos^2(x) ?
The integral of 1/x*cos^2(x) is 1/2 (ln|x|+\Ci(2x))+C