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

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

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

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

d (ln ∣ tan (x) + sec (x) ∣ + sin (x)) + C

Solution steps

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

Take the constant out:

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

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

Expand

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

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

Apply the Sum Rule:

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

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

\int \frac{1}{cos ( x )} d x = ln ∣ tan (x) + sec (x) ∣

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

= d (ln ∣ tan (x) + sec (x) ∣ + sin (x))

Add a constant to the solution

= d (ln ∣ tan (x) + sec (x) ∣ + sin (x)) + C

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