Question

Evaluate the integrals in Exercises 53–76.71. ∫(from -π/2 to π/2) 2cosθ dθ/(1+(sinθ)²)

Accepted Answer

Identify the integral to evaluate: $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{2 \cos 	heta}{1 + (\sin \$$theta)^2$$} \, d	heta. $$Consider the symmetry of the integrand. Note that $$\cos 	heta is an $$even function and $$(\sin \$$theta)^2$$ is also even, so the entire integrand is an even function. This allows us to rewrite the integral as 2$$ \$$int_0$$^{\frac{\pi}{2}} \frac{2 \cos 	heta}{1 + (\sin \$$theta)^2$$} \, d	heta$$. Use the substitution u$$ = \sin 	heta$$, which implies du$$ = \cos 	heta \, d	heta$$. This substitution will simplify the integral because the denominator is in terms of $$\sin 	heta$$ and the numerator contains $$\cos 	heta \, d	heta$$. Rewrite the integral in terms of u$: the limits change from $$	heta = 0$$ to $$	heta = \frac{\pi}{2}$$, which correspond to u = 0$ to u = 1$. The integral becomes 2$$ \$$int_0^1$$ \frac{2}{1 + $$u^2$$} \, du$$. Recognize that $$\int \frac{1}{1 + $$u^2$$} \, du$$ is the standard arctangent integral. Set up the integral accordingly and prepare to evaluate it using the antiderivative $$\arctan u$$.

Evaluate the integrals in Exercises 53–76.
71. ∫(from -π/2 to π/2) 2cosθ dθ/(1+(sinθ)²)

Verified video answer for a similar problem:

Key Concepts

Definite Integrals

Trigonometric Functions and Identities

Symmetry in Integrals