Question

Evaluate the integral. $$\int_{0}^{x / 4} \frac{1+\cos ^{2} \theta}{\cos ^{2} \theta} d \theta$$

Name: Problem 25, Chapter 5: Calculus
Uploaded: 2021-12-01T07:13:32-08:00
Duration: 1 min 58 s
Channel: Harmender Singh Yadav
Description: Evaluate the integral. ∫0^x / 4 (1+cos^2 θ)/(cos^2 θ) d θ

   Evaluate the integral.
$$\int_{0}^{x / 4} \frac{1+\cos ^{2} \theta}{\cos ^{2} \theta} d \theta$$

Calculus

James Stewart 4th Edition

Chapter 5, Problem 25

Instant Answer

Solved by Expert Harmender Singh Yadav

12/01/2021

Step 1

Step 1: First, we rewrite the integral by splitting the fraction into two parts: $$\int_{0}^{x / 4} \frac{1+\cos ^{2} \theta}{\cos ^{2} \theta} d \theta = \int_{0}^{x / 4} \frac{1}{\cos ^{2} \theta} d \theta + \int_{0}^{x / 4} \frac{\cos ^{2} \theta}{\cos ^{2} Show more…

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Evaluate the integral. $$\int_{0}^{x / 4} \frac{1+\cos ^{2} \theta}{\cos ^{2} \theta} d \theta$$

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Key Concepts

Definite Integration

Definite integration involves applying the Fundamental Theorem of Calculus, which connects antiderivatives with evaluation over a closed interval. This includes finding an antiderivative of the integrand and then subtracting its value at the lower limit from its value at the upper limit.

Integration of Trigonometric Functions

Integrating trigonometric functions often relies on standard antiderivatives, such as the fact that the antiderivative of sec²? is tan?. Recognizing these standard forms allows one to directly compute integrals without resorting to more complex methods.

Trigonometric Identities

Using trigonometric identities to simplify an integrand is a fundamental technique. In many cases, expressions involving trigonometric functions can be rewritten using identities such as the reciprocal identities (e.g., secant as the reciprocal of cosine) to transform the integral into a sum of simpler parts that are easier to integrate.

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