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

The integral of (cos(ln(x^2)))/x is 1/2 sin(2ln(x))+C

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integral of (cos(ln(x^2)))/x

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

\frac{1}{2} sin (2 ln (x)) + C

Solution steps

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

\frac{cos ( ln ( x ^{2} ) )}{x} = cos (ln (x^{2})) \frac{1}{x}

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

Apply u-substitution

: \int \frac{cos ( ln ( u ) )}{2 u} d u

= \int \frac{cos ( ln ( u ) )}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{cos ( ln ( u ) )}{u} d u

Apply u-substitution

: \int cos (v) d v

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

Use the common integral:

\int cos (v) d v = sin (v)

= \frac{1}{2} sin (v)

Substitute back

= \frac{1}{2} sin (ln (x^{2}))

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= \frac{1}{2} sin (2 ln (x))

Add a constant to the solution

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

\int \frac{( 1 + x )}{( x ^{2} - 2 x - 3 ) ^{3}} d x

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

\int \frac{ln ( ln ( 4 x ) )}{x ln ( 4 x )} d x

\int 10 - 4 x + 4 x^{2} d x

\int \frac{1}{1 - \frac{x ^{2}}{49}} d x

What is the integral of (cos(ln(x^2)))/x ?
The integral of (cos(ln(x^2)))/x is 1/2 sin(2ln(x))+C