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

The integral of (tagx)/(cos^2(x)) is atgxtan(x)+atgln|cos(x)|+C

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

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

a t gx tan (x) + a t g ln ∣ cos (x) ∣ + C

Solution steps

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

Take the constant out:

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

= a t g \cdot \int \frac{x}{cos ^{2} ( x )} d x

Rewrite using trig identities

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

Take the constant out:

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

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

Apply Integration By Parts

= a t g \cdot 2 (\frac{1}{2} x tan (x) - \int \frac{1}{2} tan (x) d x)

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

= a t g \cdot 2 (\frac{1}{2} x tan (x) - (- \frac{1}{2} ln ∣ cos (x) ∣))

Simplify

a t g \cdot 2 (\frac{1}{2} x tan (x) - (- \frac{1}{2} ln ∣ cos (x) ∣)) : a t gx tan (x) + a t g ln ∣ cos (x) ∣

= a t gx tan (x) + a t g ln ∣ cos (x) ∣

Add a constant to the solution

= a t gx tan (x) + a t g ln ∣ cos (x) ∣ + C

What is the integral of (tagx)/(cos^2(x)) ?
The integral of (tagx)/(cos^2(x)) is atgxtan(x)+atgln|cos(x)|+C