What is the integral of sec^2(2/3 x) ?

The integral of sec^2(2/3 x) is 3/2 tan((2x)/3)+C

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integral of sec^2(2/3 x)

\int sec^{2} (\frac{2}{3} x) d x

\frac{3}{2} tan (\frac{2 x}{3}) + C

Solution steps

\int sec^{2} (\frac{2}{3} x) d x

Rewrite using trig identities

= \int \frac{1}{cos ^{2} ( \frac{2 x}{3} )} d x

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{2} (\frac{2 x}{3}) d x

Apply u-substitution

= \int sec^{2} (u) \frac{3}{2} d u

Take the constant out:

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

= \frac{3}{2} \cdot \int sec^{2} (u) d u

Use the common integral:

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

= \frac{3}{2} tan (u)

Substitute back

u = \frac{2 x}{3}

= \frac{3}{2} tan (\frac{2 x}{3})

Add a constant to the solution

= \frac{3}{2} tan (\frac{2 x}{3}) + C

What is the integral of sec^2(2/3 x) ?
The integral of sec^2(2/3 x) is 3/2 tan((2x)/3)+C