- #1
1MileCrash
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Find the integral of
X^3sin(x^2)
No useful substitutions I can find, integration by parts gets chaotic.
X^3sin(x^2)
No useful substitutions I can find, integration by parts gets chaotic.
Practice, practice, practice ...1MileCrash said:Got the answer. Thank you.
I wish I were able to recognize these things more quickly.
This is a mathematical expression that represents the process of finding the antiderivative of the function x^3sin(x^2), without using any substitutions or introducing chaos into the solution.
To integrate this function without substitutions, you can use the integration by parts method. This involves splitting the original function into two parts, u and dv, and using the formula ∫u dv = uv - ∫v du. In this case, u = x^3 and dv = sin(x^2).
The phrase "No Chaos" in this context means that we are not introducing any unnecessary or complicated steps in the integration process. This allows us to focus on the fundamental principles of integration and arrive at a simpler and more precise solution.
First, we use the formula ∫u dv = uv - ∫v du to split the original function into two parts, u = x^3 and dv = sin(x^2). Next, we find the antiderivative of dv, which is ∫dv = -cos(x^2). Then, we differentiate u to find du = 3x^2 dx and integrate to find ∫u du = x^3. Finally, we use the formula to find the overall antiderivative: ∫x^3sin(x^2) dx = -x^3cos(x^2) + ∫3x^2cos(x^2) dx.
Being able to integrate functions without substitutions is important because it allows us to understand the fundamental principles of integration and develop problem-solving skills. It also helps us simplify complex expressions and equations, which is crucial in many areas of science, engineering, and mathematics.