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

The integral of x/(cos(x^2)) is 1/2 ln|tan(x^2)+sec(x^2)|+C

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

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

\frac{1}{2} ln tan (x^{2}) + sec (x^{2}) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 cos ( u )} d u

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

Rewrite using trig identities

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

Use the common integral:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = x^{2}

= \frac{1}{2} ln tan (x^{2}) + sec (x^{2})

Add a constant to the solution

= \frac{1}{2} ln tan (x^{2}) + sec (x^{2}) + C

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