Evaluate [tex] \int \sin ^2 x \cos x dx [/tex]
Answer (2)
To evaluate the integral ∫ sin 2 x cos x d x , we use u-substitution with u = sin x . After substituting and integrating, we find the result is 3 s i n 3 x + C .
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Recognize the integral as a candidate for u-substitution.
Substitute u = sin x , so d u = cos x d x , transforming the integral to ∫ u 2 d u .
Integrate to get 3 u 3 + C .
Substitute back to obtain the final answer: 3 sin 3 x .
Explanation
Problem Analysis We are asked to evaluate the indefinite integral ∫ sin 2 x cos x d x . This means we need to find a function whose derivative is sin 2 x cos x . We will also need to include the constant of integration, denoted by + C .
U-Substitution To solve this integral, we can use a simple u-substitution. Let's set u = sin x . Then, the derivative of u with respect to x is d x d u = cos x . This means that d u = cos x d x .
Substituting u and du Now we can substitute u and d u into the original integral: ∫ sin 2 x cos x d x = ∫ u 2 d u
Integrating with respect to u Next, we find the antiderivative of u 2 with respect to u . Using the power rule for integration, we have: ∫ u 2 d u = 2 + 1 u 2 + 1 + C = 3 u 3 + C
Substituting back for x Finally, we substitute back u = sin x to express the result in terms of x : 3 u 3 + C = 3 sin 3 x + C
Final Result Therefore, the indefinite integral of sin 2 x cos x is 3 s i n 3 x + C .
Examples
Imagine you're designing a curved ramp for a skateboard park. The rate of change of the ramp's slope is described by the function sin 2 ( x ) cos ( x ) . To find the actual slope of the ramp at any point, you need to integrate this function. The result, 3 s i n 3 ( x ) + C , gives you a formula to calculate the slope, ensuring a smooth and safe transition for skateboarders. Understanding integration helps you design real-world structures with precision.
