Calcule as integrais usando funções Gamma e Beta
\int_{0}^{\frac{\pi}{2}}(1+\sin \theta)^{2} \cos ^{2} \theta d \theta

Passo 1

O enunciado pede para calcular
\int_{0}^{\frac{\pi}{2}}(1+\sin\theta)^{2}\cos^{2}\theta\,d\theta
usando as funções Beta e Gamma. Vamos expandir o quadrado e dividir a integral em partes simples.

Passo 2

Expandindo (1+\sin\theta)^{2} obtemos
(1+\sin\theta)^{2}\cos^{2}\theta =\bigl(1+2\sin\theta+\sin^{2}\theta\bigr)\cos^{2}\theta
e, assim,
\int_{0}^{\frac{\pi}{2}}(1+\sin\theta)^{2}\cos^{2}\theta\,d\theta =\int_{0}^{\frac{\pi}{2}}\cos^{2}\theta\,d\theta +2\int_{0}^{\frac{\pi}{2}}\sin\theta\,\cos^{2}\theta\,d\theta +\int_{0}^{\frac{\pi}{2}}\sin^{2}\theta\,\cos^{2}\theta\,d\theta.

Passo 3

Vamos calcular
I_{1}=\int_{0}^{\frac{\pi}{2}}\cos^{2}\theta\,d\theta.
Usando a fórmula geral
\int_{0}^{\frac{\pi}{2}}\cos^{n}\theta\,d\theta =\frac{\sqrt\pi}{2}\,\frac{\Gamma\bigl(\frac{n+1}{2}\bigr)}{\Gamma\bigl(\frac{n}{2}+1\bigr)},
para n=2 vem
I_{1} =\frac{\sqrt\pi}{2}\,\frac{\Gamma\bigl(\tfrac{3}{2}\bigr)}{\Gamma(2)} =\frac{\sqrt\pi}{2}\,\frac{\tfrac{\sqrt\pi}{2}}{1} =\frac{\pi}{4}.

Passo 4

Agora
I_{2}=2\int_{0}^{\frac{\pi}{2}}\sin\theta\,\cos^{2}\theta\,d\theta.
Use a substituição t=\cos\theta, dt=-\sin\theta\,d\theta. Então
\int_{0}^{\frac{\pi}{2}}\sin\theta\,\cos^{2}\theta\,d\theta =\int_{t=1}^{t=0}t^{2}(-dt) =\int_{0}^{1}t^{2}\,dt =\frac{1}{3}.
Multiplicando por 2 dá
I_{2}=\frac{2}{3}.

Passo 5

Por fim,
I_{3}=\int_{0}^{\frac{\pi}{2}}\sin^{2}\theta\,\cos^{2}\theta\,d\theta.
Reconhecemos a forma
\int_{0}^{\frac{\pi}{2}}\sin^{2a-1}\theta\,\cos^{2b-1}\theta\,d\theta =\frac{1}{2}B(a,b),
com 2a-1=2 e 2b-1=2, ou seja a=b=\tfrac{3}{2}. Logo
I_{3} =\frac{1}{2}B\Bigl(\tfrac{3}{2},\tfrac{3}{2}\Bigr) =\frac{1}{2}\,\frac{\Gamma(\tfrac{3}{2})^{2}}{\Gamma(3)} =\frac{1}{2}\,\frac{\bigl(\tfrac{\sqrt\pi}{2}\bigr)^{2}}{2} =\frac{\pi}{16}.

Resposta

Somando tudo:
\int_{0}^{\frac{\pi}{2}}(1+\sin\theta)^{2}\cos^{2}\theta\,d\theta =I_{1}+I_{2}+I_{3} =\frac{\pi}{4}+\frac{2}{3}+\frac{\pi}{16} =\frac{15\pi+32}{48}.

Exercícios relacionados