What is the integral of 5/(sin^2(x))-4/(cos^2(x)) ?

The integral of 5/(sin^2(x))-4/(cos^2(x)) is -5cot(x)-4tan(x)+C

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integral of 5/(sin^2(x))-4/(cos^2(x))

\int \frac{5}{sin ^{2} ( x )} - \frac{4}{cos ^{2} ( x )} d x

- 5 cot (x) - 4 tan (x) + C

Solution steps

\int \frac{5}{sin ^{2} ( x )} - \frac{4}{cos ^{2} ( x )} 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{5}{sin ^{2} ( x )} d x - \int \frac{4}{cos ^{2} ( x )} d x

\int \frac{5}{sin ^{2} ( x )} d x = - 5 cot (x)

\int \frac{4}{cos ^{2} ( x )} d x = 4 tan (x)

= - 5 cot (x) - 4 tan (x)

Add a constant to the solution

= - 5 cot (x) - 4 tan (x) + C

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What is the integral of 5/(sin^2(x))-4/(cos^2(x)) ?
The integral of 5/(sin^2(x))-4/(cos^2(x)) is -5cot(x)-4tan(x)+C