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

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

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

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

x tan (x) + ln ∣ cos (x) ∣ + C

Solution steps

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

Simplify

x \frac{1}{cos ^{2} ( x )} : x sec^{2} (x)

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

Apply Integration By Parts

= x tan (x) - \int tan (x) d x

\int tan (x) d x = - ln ∣ cos (x) ∣

= x tan (x) - (- ln ∣ cos (x) ∣)

Simplify

= x tan (x) + ln ∣ cos (x) ∣

Add a constant to the solution

= x tan (x) + ln ∣ cos (x) ∣ + C

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