What is the integral of (cos(ln(4)x^2))/x ?

The integral of (cos(ln(4)x^2))/x is 1/2 \Ci(ln(4)x^2)+C

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integral of (cos(ln(4)x^2))/x

\int \frac{cos ( ln ( 4 ) x ^{2} )}{x} d x

\frac{1}{2} Ci (ln (4) x^{2}) + C

Solution steps

\int \frac{cos ( ln ( 4 ) x ^{2} )}{x} d x

Apply u-substitution

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

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

Use the nonelementary integral:

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

= \frac{1}{2} Ci (u)

Substitute back

u = ln (4) x^{2}

= \frac{1}{2} Ci (ln (4) x^{2})

Add a constant to the solution

= \frac{1}{2} Ci (ln (4) x^{2}) + C

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What is the integral of (cos(ln(4)x^2))/x ?
The integral of (cos(ln(4)x^2))/x is 1/2 \Ci(ln(4)x^2)+C