Tough Integration

Level 2

Integrate this: -sin^2(x)/cos(x) dx

-tan(x) -cot(x) -sec(x)+cos(x) cos(x)+sex(x)

Gaurav Chauhan by Gaurav Chauhan , 25, India

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1 solution

Quest Keeper
Nov 22, 2020

∫ − sin ⁡ 2 x cos ⁡ x d x = ∫ − ( 1 − cos ⁡ 2 x ) cos ⁡ x d x = ∫ cos ⁡ 2 x − 1 cos ⁡ x d x = ∫ ( cos ⁡ x − 1 cos ⁡ x ) d x = ∫ ( cos ⁡ x − sec ⁡ x ) d x \int \frac{-\sin^{2}x}{\cos x} \: dx \\ = \int \frac{-(1 - \cos^{2}x)}{\cos x} \: dx \\ = \int \frac {\cos^{2}x - 1}{\cos x} \: dx \\ = \int (\cos x - \frac {1}{\cos x}) \: dx \\ = \int (\cos x - \sec x) \: dx

As you can see, the solution provided is not the integration result, but rather the other form of the integrant. \text{As you can see, the solution provided is not the integration result, but rather the other form of the integrant.}

= ∫ ( cos ⁡ x − sec ⁡ x ) d x = ∫ cos ⁡ x d x − ∫ sec ⁡ x d x = sin ⁡ x − ln ⁡ ∣ tan ⁡ x + sec ⁡ x ∣ + c = \int (\cos x - \sec x) \: dx \\ = \int \cos x \: dx - \int \sec x \: dx \\ = \sin x - \ln |\tan x + \sec x| + c

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