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

The integral of t/(cos^2(t)) is ttan(t)+ln|cos(t)|+C

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

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

t tan (t) + ln ∣ cos (t) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

= 2 \cdot \int \frac{t}{1 + cos ( 2 t )} d t

Apply Integration By Parts

: \frac{1}{2} t tan (t) - \int \frac{1}{2} tan (t) d t

= 2 (\frac{1}{2} t tan (t) - \int \frac{1}{2} tan (t) d t)

\int \frac{1}{2} tan (t) d t = - \frac{1}{2} ln ∣ cos (t) ∣

= 2 (\frac{1}{2} t tan (t) - (- \frac{1}{2} ln ∣ cos (t) ∣))

Simplify

2 (\frac{1}{2} t tan (t) - (- \frac{1}{2} ln ∣ cos (t) ∣)) : t tan (t) + ln ∣ cos (t) ∣

= t tan (t) + ln ∣ cos (t) ∣

Add a constant to the solution

= t tan (t) + ln ∣ cos (t) ∣ + C

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What is the integral of t/(cos^2(t)) ?
The integral of t/(cos^2(t)) is ttan(t)+ln|cos(t)|+C