To simplify the integral, we use a substitution. Let t = sin θ - cos θ. This choice is motivated by the numerator (sin θ + cos θ) dθ and the denominator containing sin 2θ.

Differentiating t = sin θ - cos θ with respect to θ gives dt = (cos θ + sin θ) dθ. Squaring t allows us to express sin 2θ in terms of t, since sin² θ + cos² θ = 1 and 2 sin θ cos θ = sin 2θ.

Differentiating t = sin θ - cos θ with respect to θ gives dt = (cos θ + sin θ) dθ. Squaring t allows us to express sin 2θ in terms of t, since sin² θ + cos² θ = 1 and 2 sin θ cos θ = sin 2θ.

We need to change the limits of integration according to our substitution. When θ = 0, t = sin 0 - cos 0 = -1. When θ = (π)/(2), t = sin (π)/(2) - cos (π)/(2) = 1.

We use the standard integral formula ∫ (dx)/(a² - x²) = (1)/(2a) ln | (a + x)/(a - x) | + C. Here, a = (5)/(4). After applying the formula and evaluating at the limits, we simplify the logarithmic terms.

We use the standard integral formula ∫ (dx)/(a² - x²) = (1)/(2a) ln | (a + x)/(a - x) | + C. Here, a = (5)/(4). After applying the formula and evaluating at the limits, we simplify the logarithmic terms.

The integral $80 \int_0^{\frac{\pi}{2}} \left(\frac{\sin \theta + \cos \theta}{9 + 16 \sin 2 \theta}\right) d\theta$ is equal to:

Answer: $8 \ln 3$

Step-by-step solution

Step 1: Substitute $t = \sin \theta - \cos \theta$

To simplify the integral, we use a substitution. Let $t = \sin \theta - \cos \theta$ . This choice is motivated by the numerator $(\sin \theta + \cos \theta) d\theta$ and the denominator containing $\sin 2\theta$ .

Step 2: Find $dt$ and express $\sin 2\theta$ in terms of $t$

Differentiating $t = \sin \theta - \cos \theta$ with respect to $\theta$ gives $dt = (\cos \theta + \sin \theta) d\theta$ . Squaring $t$ allows us to express $\sin 2\theta$ in terms of $t$ , since $\sin^2 \theta + \cos^2 \theta = 1$ and $2 \sin \theta \cos \theta = \sin 2\theta$ .

Step 3: Change the limits of integration

We need to change the limits of integration according to our substitution. When $\theta = 0$ , $t = \sin 0 - \cos 0 = -1$ . When $\theta = \frac{\pi}{2}$ , $t = \sin \frac{\pi}{2} - \cos \frac{\pi}{2} = 1$ .

Step 4: Rewrite the integral in terms of $t$

Substitute $dt$ and $1 - t^2$ 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 $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a + x}{a - x} \right| + C$ . Here, $a = \frac{5}{4}$ . After applying the formula and evaluating at the limits, we simplify the logarithmic terms.

Step 6: Simplify the result

Using the logarithm property $\ln(a^b) = b \ln a$ , we can simplify $4 \ln 9$ to $8 \ln 3$ .

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The integral 80∫0π2(sin⁡θ+cos⁡θ9+16sin⁡2θ)dθ80 \int_0^{\frac{\pi}{2}} \left(\frac{\sin \theta + \cos \theta}{9 + 16 \sin 2 \theta}\right) d\theta80∫02π​​(9+16sin2θsinθ+cosθ​)dθ is equal to: