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

The integral of (2x)/(cos^2(x)) is 4(1/2 xtan(x)+1/2 ln|cos(x)|)+C

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

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

4 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣) + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= 2 \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

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

Apply Integration By Parts

= 2 \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) ∣

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

Simplify

= 4 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣)

Add a constant to the solution

= 4 (\frac{1}{2} x tan (x) + \frac{1}{2} ln ∣ cos (x) ∣) + C

What is the integral of (2x)/(cos^2(x)) ?
The integral of (2x)/(cos^2(x)) is 4(1/2 xtan(x)+1/2 ln|cos(x)|)+C