What is the integral of (tan(x/3)+cot(x/3))^2 ?

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

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integral of (tan(x/3)+cot(x/3))^2

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

3 tan (\frac{x}{3}) - 3 cot (\frac{x}{3}) + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{( sin ^{2} ( \frac{x}{3} ) + cos ^{2} ( \frac{x}{3} ) ) ^{2}}{cos ^{2} ( \frac{x}{3} ) sin ^{2} ( \frac{x}{3} )} : \frac{sin ^{2} ( \frac{x}{3} )}{cos ^{2} ( \frac{x}{3} )} + 2 + \frac{cos ^{2} ( \frac{x}{3} )}{sin ^{2} ( \frac{x}{3} )}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{sin ^{2} ( \frac{x}{3} )}{cos ^{2} ( \frac{x}{3} )} d x = - x + 3 tan (\frac{x}{3})

\int 2 d x = 2 x

\int \frac{cos ^{2} ( \frac{x}{3} )}{sin ^{2} ( \frac{x}{3} )} d x = - x - 3 cot (\frac{x}{3})

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

Simplify

= 3 tan (\frac{x}{3}) - 3 cot (\frac{x}{3})

Add a constant to the solution

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

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