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

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

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integral of 2/3 sec^2(x/3)

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = \frac{x}{3}

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

Simplify

\frac{2}{3} \cdot 3 tan (\frac{x}{3}) : 2 tan (\frac{x}{3})

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

Add a constant to the solution

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

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