What is the integral of 1/(cos^2(2x-3)) ?

The integral of 1/(cos^2(2x-3)) is 1/2 tan(2x-3)+C

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integral of 1/(cos^2(2x-3))

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

\frac{1}{2} tan (2 x - 3) + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

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

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

Substitute back

u = 2 x - 3

= \frac{1}{2} tan (2 x - 3)

Add a constant to the solution

= \frac{1}{2} tan (2 x - 3) + C

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What is the integral of 1/(cos^2(2x-3)) ?
The integral of 1/(cos^2(2x-3)) is 1/2 tan(2x-3)+C