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

The integral of ((sin(t)+cos(t))^2)/(cos(t)) is ln|tan(t)+sec(t)|-2cos(t)+C

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

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

ln ∣ tan (t) + sec (t) ∣ - 2 cos (t) + C

Solution steps

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

Expand

\frac{( sin ( t ) + cos ( t ) ) ^{2}}{cos ( t )} : \frac{sin ^{2} ( t )}{cos ( t )} + 2 sin (t) + cos (t)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{sin ^{2} ( t )}{cos ( t )} d t = ln ∣ tan (t) + sec (t) ∣ - sin (t)

\int 2 sin (t) d t = - 2 cos (t)

\int cos (t) d t = sin (t)

= ln ∣ tan (t) + sec (t) ∣ - sin (t) - 2 cos (t) + sin (t)

Simplify

= ln ∣ tan (t) + sec (t) ∣ - 2 cos (t)

Add a constant to the solution

= ln ∣ tan (t) + sec (t) ∣ - 2 cos (t) + C

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What is the integral of ((sin(t)+cos(t))^2)/(cos(t)) ?
The integral of ((sin(t)+cos(t))^2)/(cos(t)) is ln|tan(t)+sec(t)|-2cos(t)+C