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

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

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integral of 1/(cos^2(r)x)

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

sec^{2} (r) ln ∣ x ∣ + C

Solution steps

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{x} d x = ln (∣ x ∣)

= \frac{1}{cos ^{2} ( r )} ln ∣ x ∣

Use the basic trigonometric identity:

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

= sec^{2} (r) ln ∣ x ∣

Add a constant to the solution

= sec^{2} (r) ln ∣ x ∣ + C

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