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

The integral of (2sin(x))/(cos^2(x)) is 2sec(x)+C

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

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

2 sec (x) + C

Solution steps

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

Apply u-substitution

= 2 \cdot \int - \frac{1}{u ^{2}} d u

Take the constant out:

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

= 2 (- \int \frac{1}{u ^{2}} d u)

Apply the Power Rule

= 2 (- (- \frac{1}{u}))

Substitute back

u = cos (x)

= 2 (- (- \frac{1}{cos ( x )}))

Simplify

2 (- (- \frac{1}{cos ( x )})) : 2 sec (x)

= 2 sec (x)

Add a constant to the solution

= 2 sec (x) + C

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What is the integral of (2sin(x))/(cos^2(x)) ?
The integral of (2sin(x))/(cos^2(x)) is 2sec(x)+C