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

The integral of 1/(2cos^2(x/2)) is tan(x/2)+C

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

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

tan (\frac{x}{2}) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int 2 sec^{2} (u) d u

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

Take the constant out:

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

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

Use the common integral:

\int sec^{2} (u) d u = tan (u)

= \frac{1}{2} \cdot 2 tan (u)

Substitute back

u = \frac{x}{2}

= \frac{1}{2} \cdot 2 tan (\frac{x}{2})

Simplify

\frac{1}{2} \cdot 2 tan (\frac{x}{2}) : tan (\frac{x}{2})

= tan (\frac{x}{2})

Add a constant to the solution

= tan (\frac{x}{2}) + C

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