Question

Use any method to evaluate the integrals in Exercises 65–70.∫ cot(x) / cos²(x) dx

Accepted Answer

Rewrite the integrand to express it in terms of sine and cosine functions. Recall that $$\cot(x) = \frac{\cos(x)}{\sin(x)}$$, so the integral becomes $$\int \frac{\cot(x)}{\cos$$^{2}$$(x)} \, dx = \int \frac{\cos(x)}{\sin(x) \cos$$^{2}$$(x)} \, dx. $$Simplify the integrand by canceling one $$\cos(x)$$ term in the numerator and denominator, resulting in $$\int \frac{1}{\sin(x) \cos(x)} \, dx. $$Consider a substitution to simplify the integral. One useful substitution is to let $$u = \sin(x)$$, which implies $$du = \cos(x) \, dx. $$Rearranging, we get $$dx = \frac{du}{\cos(x)}. $$Substitute $$u$$ and $$dx$$ back into the integral. The integral becomes $$\int \frac{1}{u \cos(x)} \cdot \frac{du}{\cos(x)} = \int \frac{1}{u \cos$$^{2}$$(x)} \, du. $$Since this still contains $$\cos(x)$$, consider an alternative substitution or rewrite the integrand differently. Alternatively, rewrite the original integrand as $$\cot(x) \sec$$^{2}$$(x)$$ because $$\frac{1}{\cos$$^{2}$$(x)} = \sec$$^{2}$$(x). $$Then, use the substitution $$u = \sin(x)$$, so $$du = \cos(x) \, dx. $$Express $$\cot(x) \sec$$^{2}$$(x) \, dx in $$terms of $$u$$ and $$du to $$evaluate the integral.

Use any method to evaluate the integrals in Exercises 65–70.
∫ cot(x) / cos²(x) dx

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Key Concepts

Trigonometric Identities

Integration Techniques

Handling Rational Trigonometric Functions