Skip to content
Notely Maths 12
Class 12 Maths
🚨 This site is a work in progress. Exciting updates are coming soon!

3.6 Q-21

Question Statement

Evaluate the following integral:

∫0π4Sec⁡θSin⁡θ+Cos⁡θ,dθ\int_{0}^{\frac{\pi}{4}} \frac{\operatorname{Sec} \theta}{\operatorname{Sin} \theta + \operatorname{Cos} \theta} , d\theta

Background and Explanation

In this problem, we are asked to solve a definite integral involving the secant and trigonometric functions sine and cosine. To simplify, we’ll use some key trigonometric identities and manipulation techniques.

Key Concepts:

  • Secant and Tangent: Recall that sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} and tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.
  • Logarithmic Integration: When you encounter expressions that simplify into a fraction involving trigonometric functions, you may be able to use logarithmic properties to solve the integral.

Solution

Let’s break the solution down step by step.

Step 1: Simplify the integral

Start by manipulating the integral’s expression:

∫0π4Sec⁡θSin⁡θ+Cos⁡θ,dθ\int_{0}^{\frac{\pi}{4}} \frac{\operatorname{Sec} \theta}{\operatorname{Sin} \theta + \operatorname{Cos} \theta} , d\theta

We can express secant as sec⁡θ=1cos⁡θ \sec \theta = \frac{1}{\cos \theta}. So the integral becomes:

=∫0π41cos⁡θsin⁡θ+cos⁡θ,dθ= \int_{0}^{\frac{\pi}{4}} \frac{\frac{1}{\cos \theta}}{\sin \theta + \cos \theta} , d\theta

Step 2: Use a trigonometric identity

Notice that the denominator sin⁡θ+cos⁡θ\sin \theta + \cos \theta can be rewritten. Factor out the cos⁡θ\cos \theta from the denominator:

=∫0π4Sec⁡2θ(tan⁡θ+1),dθ= \int_{0}^{\frac{\pi}{4}} \frac{\operatorname{Sec}^2 \theta}{(\tan \theta + 1)} , d\theta

This simplifies the integral into a form that can be easier to handle.

Step 3: Use substitution and simplify

Now, we recognize that sec⁡2θ,dθ\sec^2 \theta , d\theta is the derivative of tan⁡θ\tan \theta, which leads us to the following:

=∫0π41(tan⁡θ+1),d(tan⁡θ)= \int_{0}^{\frac{\pi}{4}} \frac{1}{(\tan \theta + 1)} , d(\tan \theta)

This is a standard integral that can be solved using the natural logarithm. We apply the formula:

∫1u,du=ln⁡∣u∣\int \frac{1}{u} , du = \ln |u|

where u=tan⁡θ+1u = \tan \theta + 1.

Thus, the integral becomes:

=ln⁡(1+tan⁡θ)∣0π4= \ln \left(1 + \tan \theta \right) \Bigg|_0^{\frac{\pi}{4}}

Step 4: Evaluate the integral

Now, evaluate the integral at the limits of integration, θ=0\theta = 0 and θ=π4\theta = \frac{\pi}{4}:

=ln⁡(1+tan⁡π4)−ln⁡(1+tan⁡0)= \ln \left(1 + \tan \frac{\pi}{4}\right) - \ln \left(1 + \tan 0\right)

Since tan⁡π4=1\tan \frac{\pi}{4} = 1 and tan⁡0=0\tan 0 = 0, this simplifies to:

=ln⁡(1+1)−ln⁡(1+0)=ln⁡2−ln⁡1= \ln (1 + 1) - \ln (1 + 0) = \ln 2 - \ln 1

And since ln⁡1=0\ln 1 = 0, we get:

=ln⁡2= \ln 2

Key Formulas or Methods Used

  • Trigonometric Identities: sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.
  • Integration by Substitution: Used to simplify the integral into a form involving a logarithm.
  • Logarithmic Integration: The integral of 1x\frac{1}{x} is ln⁡∣x∣\ln |x|.

Summary of Steps

  1. Rewrite the integral using the identity sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}.
  2. Simplify the denominator by factoring out cos⁡θ\cos \theta and using the identity for tan⁡θ\tan \theta.
  3. Apply substitution to transform the integral into a logarithmic form.
  4. Evaluate the integral by applying the limits and simplifying the result.

The final answer is:

ln⁡2\ln 2