int_(0)^((pi)/(4))sin^(3)(x)*cos(x)dx

Given Integral: We are given the integral to evaluate: int_(0)^((pi)/(4))sin^(3)(x)*cos(x)dx To solve this integral, we can use the substitution method. Let's choose u = sin(x), which means du = cos(x)dx.Substitution Method: Now we differentiate u with respect to x to find du: du/dx = cos(x) du = cos(x)dxDifferentiate u: Substitute sin(x) with u and cos(x)dx with du in the integral: int_(0)^((pi)/(4))sin^(3)(x)*cos(x)dx = int_(sin(0))^((sin(pi)/(4)))u^3*duSubstitute u and du: We need to change the limits of integration because we changed the variable of integration from x to u. When x = 0, u = sin(0) = 0. When x = π/4, u = sin(π/4) = √2/2. So the new limits of integration are from 0 to √2/2.Change Limits of Integration: Now we can integrate u^3 with respect to u: int_(0)^((√2)/(2))u^3 du = [u^4/4]_(0)^((√2)/(2))Integrate u^3: Evaluate the antiderivative at the upper and lower limits: [u^4/4]_(0)^((√2)/(2)) = ((√2/2)^4)/4 - (0^4)/4Evaluate Antiderivative: Simplify the expression: ((√2/2)^4)/4 = (2^2/2^4)/4 = 1/8 (0^4)/4 = 0 So the result is 1/8 - 0 = 1/8.Simplify Expression: The final answer is the value of the definite integral: int_(0)^((pi)/(4))sin^(3)(x)*cos(x)dx = 1/8

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$\int_{0}^{\frac{\pi}{4}} \sin ^{3}(x) \cdot \cos (x) d x$

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\int_{0}^{\frac{\pi}{4}} \sin ^{3}(x) \cdot \cos (x) d x

Given Integral: We are given the integral to evaluate: $\newline$ $\int_{0}^{\frac{\pi}{4}}\sin^{3}(x)\cos(x)\,dx$ $\newline$ To solve this integral, we can use the substitution method. Let's choose $u = \sin(x)$ , which means $du = \cos(x)\,dx$ .
Substitution Method: Now we differentiate $u$ with respect to $x$ to find $du$ : $\frac{du}{dx} = \cos(x)$ $du = \cos(x)dx$
Differentiate $u$ : Substitute $\sin(x)$ with $u$ and $\cos(x)dx$ with $du$ in the integral: $\newline$ $\int_{0}^{\left(\frac{\pi}{4}\right)}\sin^{3}(x)\cos(x)dx = \int_{\sin(0)}^{\left(\sin\left(\frac{\pi}{4}\right)\right)}u^3\,du$
Substitute $u$ and $du$ : We need to change the limits of integration because we changed the variable of integration from $x$ to $u$ . When $x = 0$ , $u = \sin(0) = 0$ . When $x = \frac{\pi}{4}$ , $u = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ . So the new limits of integration are from $0$ to $\frac{\sqrt{2}}{2}$ .
Change Limits of Integration: Now we can integrate $u^3$ with respect to $u$ : $\newline$ $\int_{0}^{(\sqrt{2}/2)}u^3 du = \left[\frac{u^4}{4}\right]_{0}^{(\sqrt{2}/2)}$
Integrate $u^3$ : Evaluate the antiderivative at the upper and lower limits: $\newline$ $\left.\frac{u^4}{4}\right|_{0}^{\left(\frac{\sqrt{2}}{2}\right)} = \frac{\left(\frac{\sqrt{2}}{2}\right)^4}{4} - \frac{0^4}{4}$
Evaluate Antiderivative: Simplify the expression: $\newline$ (\sqrt{2}/2)^4)/4 = (2^2/2^4)/4 = 1/8$\newline\$(0^4)/4 = 0\(\newline$\)So the result is $1/8 - 0 = 1/8$.
Simplify Expression: The final answer is the value of the definite integral: $\int_{0}^{\frac{\pi}{4}}\sin^{3}(x)\cos(x)dx = \frac{1}{8}$