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

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

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integral of sec^2(x/2)

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= 2 tan (u)

Substitute back

u = \frac{x}{2}

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

Add a constant to the solution

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

f (x) = - 2 x^{3}, at x = - 1

(1, - \frac{1}{2}), (- 2, - \frac{7}{2})

\int_{0}^{\frac{π}{2}} x^{3} cos (2 x) d x

\frac{d}{d x} (\frac{tan ( 3 x ^{3} + x ^{2} )}{e ^{x}})^{3}

x \to 2 lim (- x^{2} + 2 x - 6)