Chapter 7: Integrals

Key Integration Formulas:
• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
• ∫ 1/x dx = log|x| + C
• ∫ e^x dx = e^x + C
• ∫ sin x dx = −cos x + C
• ∫ cos x dx = sin x + C
• ∫ a^x dx = a^x/log a + C
• Integration by parts: ∫ u dv = uv − ∫ v du

Q1. Find: ∫ (x² + 1)/(x + 1) dx

Solution

(x² + 1)/(x + 1) = (x² − 1 + 2)/(x + 1) = (x−1) + 2/(x+1)

∫ (x² + 1)/(x+1) dx = ∫ (x−1) dx + 2 ∫ 1/(x+1) dx
= x²/2 − x + 2 log|x + 1| + C

Q2. Find: ∫ sin²x dx

Solution

Using sin²x = (1 − cos 2x)/2:

∫ sin²x dx = ∫ (1 − cos 2x)/2 dx
= (1/2)[x − sin 2x/2] + C
= x/2 − sin 2x/4 + C

Q3. Find: ∫ x · e^x dx (using integration by parts)

Solution

Let u = x, dv = e^x dx
Then du = dx, v = e^x

∫ x e^x dx = x e^x − ∫ e^x dx
= x e^x − e^x + C
= e^x(x − 1) + C

Q4. Evaluate: ∫₀^π/2 sin³x dx

Solution

sin³x = sin x · sin²x = sin x(1 − cos²x)

∫₀^π/2 sin x(1 − cos²x) dx
Let cos x = t, −sin x dx = dt

When x = 0, t = 1; x = π/2, t = 0
= ∫₁⁰ (−1)(1 − t²) dt = ∫₀¹ (1 − t²) dt
= [t − t³/3]₀¹ = 1 − 1/3 = 2/3

Q5. Find: ∫ dx/(x² + 4x + 13)

Solution

x² + 4x + 13 = (x + 2)² + 9 = (x + 2)² + 3²

∫ dx/[(x+2)² + 3²] = (1/3) tan⁻¹[(x+2)/3] + C

Visual representation of definite integral as area under curve

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