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

The integral of (xtan(x^2))/(cos(x^2)) is 1/2 sec(x^2)+C

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

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

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

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \int x tan (x^{2}) sec (x^{2}) d x

Apply u-substitution

: \int \frac{tan ( u ) sec ( u )}{2} d u

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

Apply u-substitution

: \int 1 d v

= \frac{1}{2} \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= \frac{1}{2} \cdot 1 \cdot v

Substitute back

= \frac{1}{2} \cdot 1 \cdot sec (x^{2})

Simplify

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

Add a constant to the solution

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

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