Definite Integration
Integration using standard results
Grade 12
Question:
<p>If \(I = 98\int_0^1 (\sin 2)e^x(\tan x + \sec^2 x)\,dx\) and \(I = p \cdot e \cdot \sin^2 q\), find \(p + q\).</p>
Step-by-Step Solution
Key Concept: Recognize that the integrand contains the derivative of a product: d/dx[e^x tan x] = e^x(tan x + sec²x). Use integration by parts or direct substitution to evaluate.
<p><strong>Step 1:</strong> Recognize the integrand structure. Notice that d/dx[e^x tan x] = e^x tan x + e^x sec²x = e^x(tan x + sec²x).</p><p><strong>Step 2:</strong> Therefore: ∫₀¹ e^x(tan x + sec²x)dx = [e^x tan x]₀¹ = e¹ tan 1 - e⁰ tan 0 = e tan 1.</p><p><strong>Step 3:</strong> Calculate I: I = 98 · sin 2 · e tan 1.</p><p><strong>Step 4:</strong> Use the identity sin 2 = 2 sin 1 cos 1 and tan 1 = sin 1/cos 1. Therefore: I = 98 · 2 sin 1 cos 1 · e · (sin 1/cos 1) = 98 · 2 · e · sin²1 = 196 · e · sin²1.</p><p><strong>Step 5:</strong> Compare with I = p · e · sin²q: we get p = 196 and q = 1.</p><p>∴ Answer: p + q = 196 + 1 = <strong>197</strong></p>
Correct Answer: 197