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[ ( x^2 ) ( sinx ) ] / (1 + x^6)
jbunniii said:Don't listen to these clowns. The answer you seek is
[ ( u^2 ) ( sinu ) ] / (1 + u^6)
To perform efficient U-substitution for (x^2)(sinx)/(1+x^6), you first need to identify the part of the expression that can be substituted with a single variable, known as "u". In this case, the expression (1+x^6) can be substituted with u. Then, you need to find the derivative of u with respect to x, which is du/dx = 6x^5. Finally, you can rewrite the original expression in terms of u and du, and use the substitution formula to solve for the integral.
The substitution formula for U-substitution is: ∫f(x) dx = ∫f(u) (du/dx) dx. This formula allows you to substitute a variable, u, and its derivative, du/dx, in place of a more complex expression in order to simplify the integral.
No, U-substitution can only be used for integrals where the integrand (the function being integrated) can be rewritten in terms of a single variable and its derivative. This is why it is important to carefully choose the substitution variable, u, in order to make the integral simpler.
Yes, there are a few rules to follow when using U-substitution. First, you must make sure that the expression inside the integral can be rewritten in terms of a single variable and its derivative. Second, you must make sure to adjust the limits of integration when substituting u for the original variable. Third, you must use the substitution formula to rewrite the integral in terms of u and du. Lastly, you must remember to substitute back in the original variable at the end to get the final answer.
Yes, U-substitution can be used for both indefinite and definite integrals. However, when using U-substitution for a definite integral, it is important to adjust the limits of integration to match the substitution. This is done by substituting the original variable (x) into the limits and then replacing it with u.