Question 7 of 33beginner🔧 ApplyShort Answer2 marks

Find the following integrals in Exercises 6 to 20: ∫sec⁡\int \sec x (sec⁡\sec x + tan⁡\tan x) dx

Correct Answer

tan x + sec x + C

Exercise: EXERCISE 7.1 | Q: 18 | (Page 10)
For More Understanding

Explanation

The question requires evaluating the integral of a trigonometric function. The provided context contains standard integrals and substitution methods. Specifically, Chunk 3 implies the integral of secant squared through the substitution u = tan t, and Chunk 2 provides the integral of tan x, which can be differentiated to find the integral of sec x tan x.

Solution Steps

  1. Step 1: Expand the integrand. We have ∫sec⁡x(sec⁡x+tan⁡x)dx=∫(sec⁡2x+sec⁡xtan⁡x)dx\int \sec x (\sec x + \tan x) dx = \int (\sec^2 x + \sec x \tan x) dx.

  2. Step 2: Separate the terms. =∫sec⁡2xdx+∫sec⁡xtan⁡xdx= \int \sec^2 x dx + \int \sec x \tan x dx.

  3. Step 3: Evaluate the first integral. From the context's substitution logic where d(tan⁡t)=sec⁡2tdtd(\tan t) = \sec^2 t dt, we know ∫sec⁡2xdx=tan⁡x\int \sec^2 x dx = \tan x.

  4. Step 4: Evaluate the second integral. The context states ∫tan⁡xdx=log⁡∣sec⁡x∣\int \tan x dx = \log |\sec x|. Differentiating both sides gives tan⁡x=1sec⁡xddx(sec⁡x)\tan x = \frac{1}{\sec x} \frac{d}{dx}(\sec x), which implies ddx(sec⁡x)=sec⁡xtan⁡x\frac{d}{dx}(\sec x) = \sec x \tan x. Thus, ∫sec⁡xtan⁡xdx=sec⁡x\int \sec x \tan x dx = \sec x.

  5. Step 5: Combine the results. tan⁡x+sec⁡x+C\tan x + \sec x + C.

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