What is the integral of (cot(b)θ+tan(b)θ)^2 ?

The integral of (cot(b)θ+tan(b)θ)^2 is (θ^3)/(3sin^2(b)cos^2(b))+C

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integral of (cot(b)θ+tan(b)θ)^2

\int (cot (b) θ + tan (b) θ)^{2} d θ

\frac{θ ^{3}}{3 sin ^{2} ( b ) cos ^{2} ( b )} + C

Solution steps

\int (cot (b) θ + tan (b) θ)^{2} d θ

Rewrite using trig identities

= \int \frac{( θ cos ^{2} ( b ) + θ sin ^{2} ( b ) ) ^{2}}{sin ^{2} ( b ) cos ^{2} ( b )} d θ

Take the constant out:

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

= \frac{1}{sin ^{2} ( b ) cos ^{2} ( b )} \cdot \int (θ cos^{2} (b) + θ sin^{2} (b))^{2} d θ

Simplify

θ cos^{2} (b) + θ sin^{2} (b) : θ

= \frac{1}{sin ^{2} ( b ) cos ^{2} ( b )} \cdot \int θ^{2} d θ

Apply the Power Rule

= \frac{1}{sin ^{2} ( b ) cos ^{2} ( b )} \cdot \frac{θ ^{3}}{3}

Simplify

\frac{1}{sin ^{2} ( b ) cos ^{2} ( b )} \cdot \frac{θ ^{3}}{3} : \frac{θ ^{3}}{3 sin ^{2} ( b ) cos ^{2} ( b )}

= \frac{θ ^{3}}{3 sin ^{2} ( b ) cos ^{2} ( b )}

Add a constant to the solution

= \frac{θ ^{3}}{3 sin ^{2} ( b ) cos ^{2} ( b )} + C

What is the integral of (cot(b)θ+tan(b)θ)^2 ?
The integral of (cot(b)θ+tan(b)θ)^2 is (θ^3)/(3sin^2(b)cos^2(b))+C