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

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

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integral of sec^2(x/a)

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= a tan (u)

Substitute back

u = \frac{x}{a}

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

Add a constant to the solution

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

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t \to 0 lim (\frac{( tan ( t ) sec ( 2 t ) )}{3 t})

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ln (x)