Determine a integral indefinida integral sin squared invisible function application left parenthesis x right parenthesis cos invisible function application left parenthesis x right parenthesis d x. a. integral sin squared invisible function application left parenthesis x right parenthesis cos invisible function application left parenthesis x right parenthesis d x equals fraction numerator sin cubed invisible function application left parenthesis x right parenthesis over denominator 3 end fraction b. integral sin squared invisible function application left parenthesis x right parenthesis cos invisible function application left parenthesis x right parenthesis d x equals sin cubed invisible function application left parenthesis x right parenthesis c. integral sin squared invisible function application left parenthesis x right parenthesis cos invisible function application left parenthesis x right parenthesis d x equals 3 sin cubed invisible function application left parenthesis x right parenthesis d. integral sin squared invisible function application left parenthesis x right parenthesis cos invisible function application left parenthesis x right parenthesis d x equals 1 half x sin invisible function application left parenthesis x right parenthesis minus 1 fourth sin invisible function application left parenthesis x right parenthesis sin invisible function application left parenthesis 2 x right parenthesis e. integral sin squared invisible function application left parenthesis x right parenthesis cos invisible function application left parenthesis x right parenthesis d x equals 2 sin invisible function application left parenthesis x right parenthesis cos squared invisible function application left parenthesis x right parenthesis minus sin cubed invisible function application left parenthesis x right parenthesis

Question

Determine a integral indefinida integral sin squared invisible function application left parenthesis x right parenthesis cos invisible function app...

Ed · Answer

Para resolver a integral indefinida $\int \sin^2(x) \cos(x) \, dx$, podemos usar a substituição. Vamos definir $u = \sin(x)$, então $du = \cos(x) \, dx$. Assim, a integral se transforma em:

$$
\int \sin^2(x) \cos(x) \, dx = \int u^2 \, du
$$

A integral de $u^2$ é:

$$
\frac{u^3}{3} + C
$$

Substituindo $u$ de volta, temos:

$$
\frac{\sin^3(x)}{3} + C
$$

Agora, vamos analisar as alternativas:

a) $\int \sin^2(x) \cos(x) \, dx = \frac{\sin^3(x)}{3}$ - Correto!

b) $\int \sin^2(x) \cos(x) \, dx = \sin^3(x)$ - Incorreto.

c) $\int \sin^2(x) \cos(x) \, dx = 3 \sin^3(x)$ - Incorreto.

d) $\int \sin^2(x) \cos(x) \, dx = \frac{1}{2} x \sin(x) - \frac{1}{4} \sin(x) \sin(2x)$ - Incorreto.

e) $\int \sin^2(x) \cos(x) \, dx = 2 \sin(x) \cos^2(x) - \sin^3(x)$ - Incorreto.

Portanto, a alternativa correta é a) $\frac{\sin^3(x)}{3}$.

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