The integral 80∫02π(9+16sin2θsinθ+cosθ)dθ is equal to:
Get the complete, step-by-step math solution for: "The integral 80 _0^{(π)/(2)} (sin θ + cos θ)/(9 + 16 sin 2 θ) dθ is equal to:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Substitute t=sinθ−cosθ
To simplify the integral, we use a substitution. Let t=sinθ−cosθ. This choice is motivated by the numerator (sinθ+cosθ)dθ and the denominator containing sin2θ.
Step 2: Find dt and express sin2θ in terms of t
Differentiating t=sinθ−cosθ with respect to θ gives dt=(cosθ+sinθ)dθ. Squaring t allows us to express sin2θ in terms of t, since sin2θ+cos2θ=1 and 2sinθcosθ=sin2θ.
Step 3: Change the limits of integration
We need to change the limits of integration according to our substitution. When θ=0, t=sin0−cos0=−1. When θ=2π, t=sin2π−cos2π=1.
Step 4: Rewrite the integral in terms of t
Substitute dt and 1−t2 into the original integral and change the limits. This simplifies the integral to a standard form.
Step 5: Evaluate the integral
We use the standard integral formula ∫a2−x2dx=2a1lna−xa+x+C. Here, a=45. After applying the formula and evaluating at the limits, we simplify the logarithmic terms.
Step 6: Simplify the result
Using the logarithm property ln(ab)=blna, we can simplify 4ln9 to 8ln3.