Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    
                                                                                                                                                                    
  ∫₀^π/² (cos θ sin θ) / √(cos² θ + 16) dθ (Hint: Begin with u = cos θ .)

Verified step by step guidance

1. Step 1: Begin by making the substitution u = cos(θ). This implies that du = -sin(θ)dθ. Rewrite the integral in terms of u, noting that when θ = 0, u = cos(0) = 1, and when θ = π/2, u = cos(π/2) = 0.
2. Step 2: Substitute u = cos(θ) and du = -sin(θ)dθ into the integral. The integral becomes ∫₁⁰ (-u / √(u² + 16)) du. The negative sign can be used to reverse the limits of integration, changing the integral to ∫₀¹ (u / √(u² + 16)) du.
3. Step 3: To simplify further, consider a second substitution. Let v = u² + 16, which implies dv = 2u du. Rewrite the integral in terms of v, noting that when u = 0, v = 16, and when u = 1, v = 17.
4. Step 4: Substitute v = u² + 16 and dv = 2u du into the integral. The integral becomes (1/2) ∫₁⁷ (1 / √v) dv. The factor of 1/2 comes from the substitution dv = 2u du.
5. Step 5: Evaluate the integral ∫₁⁷ (1 / √v) dv using the formula for the integral of 1/√v, which is 2√v. Substitute the limits of integration (v = 16 and v = 17) into the result to complete the solution.

Key Concepts

Substitution in Integration

Substitution is a technique used in integration to simplify the integral by changing the variable of integration. By substituting a new variable, often denoted as 'u', for a function of the original variable, the integral can become easier to evaluate. This method is particularly useful when dealing with composite functions or when the integrand contains complicated expressions.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. These identities, such as sin²θ + cos²θ = 1, can be used to simplify integrals involving trigonometric functions. Understanding these identities is crucial for manipulating expressions and making substitutions in integrals that contain trigonometric terms.

Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫[a,b] f(x) dx, where 'a' and 'b' are the limits of integration. Evaluating definite integrals often involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus, which connects differentiation and integration.