Recognize Problem: Recognize the integral that needs to be solved. We need to integrate sin^3(x) with respect to x.Use Identity: Use the identity sin^2(x) = 1 - cos^2(x) to rewrite sin^3(x) as sin(x) * sin^2(x). sin^3(x) = sin(x) * (1 - cos^2(x))Substitute and Split: Substitute sin^2(x) in the integral. ∫sin^3(x)dx = ∫sin(x) * (1 - cos^2(x))dxIntegrate sin(x): Split the integral into two separate integrals. ∫sin(x) * (1 - cos^2(x))dx = ∫sin(x)dx - ∫sin(x)cos^2(x)dxUse Substitution: Integrate the first part ∫sin(x)dx. The integral of sin(x) with respect to x is -cos(x). So, ∫sin(x)dx = -cos(x)Integrate u^2: Use substitution to integrate the second part ∫sin(x)cos^2(x)dx. Let u = cos(x), then du = -sin(x)dx. Rewrite the integral: -∫u^2duSubstitute back: Integrate -∫u^2du. The integral of u^2 with respect to u is (1/3)u^3. So, -∫u^2du = -(1/3)u^3Combine Results: Substitute back cos(x) for u. -(1/3)u^3 becomes -(1/3)cos^3(x).Combine the results from Step 5 and Step 8. The final answer is the sum of the two integrals: -cos(x) - (1/3)cos^3(x) + C, where C is the constant of integration.

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\int \sin ^{3} x d x

Recognize Problem: Recognize the integral that needs to be solved. $\newline$ We need to integrate $\sin^3(x)$ with respect to $x$ .
Use Identity: Use the identity $\sin^2(x) = 1 - \cos^2(x)$ to rewrite $\sin^3(x)$ as $\sin(x) \cdot \sin^2(x)$ . $\newline$ $\sin^3(x) = \sin(x) \cdot (1 - \cos^2(x))$
Substitute and Split: Substitute $\sin^2(x)$ in the integral. $\newline$ $\int \sin^3(x)\,dx = \int \sin(x) \cdot (1 - \cos^2(x))\,dx$
Integrate $\sin(x)$ : Split the integral into two separate integrals. $\newline$ \int\sin(x) \cdot (\(1 - \cos^ $2$ (x))\,dx = \int\sin(x)\,dx - \int\sin(x)\cos^ $2$ (x)\,dx
Use Substitution: Integrate the first part $\int \sin(x)\,dx$ . The integral of $\sin(x)$ with respect to $x$ is $-\cos(x)$ . So, $\int \sin(x)\,dx = -\cos(x)$
Integrate $u^2$ : Use substitution to integrate the second part $\int \sin(x)\cos^2(x)\,dx$ . Let $u = \cos(x)$ , then $du = -\sin(x)\,dx$ . Rewrite the integral: $-\int u^2\,du$
Substitute back: Integrate $-\int u^2\,du$ . $\newline$ The integral of $u^2$ with respect to $u$ is $(1/3)u^3$ . $\newline$ So, $-\int u^2\,du = -(1/3)u^3$
Combine Results: Substitute back $\cos(x)$ for $u$ . $\newline$ $-\frac{1}{3}u^3$ becomes $-\frac{1}{3}\cos^3(x)$ .
Combine Results: Substitute back $\cos(x)$ for $u$ . $-\frac{1}{3}u^3$ becomes $-\frac{1}{3}\cos^3(x)$ .Combine the results from Step $5$ and Step $8$ . The final answer is the sum of the two integrals: $-\cos(x) - \frac{1}{3}\cos^3(x) + C$ , where $C$ is the constant of integration.