# What is #f(x) = int (x^2sinx^3- cot3x)dx# if #f(pi/12)=-1 #?

It is

#f(x)=int -((cosx^3)')/3dx-int (log(sin3x)')/3dx=>
f(x)=-1/3*cos(x^3)-1/3*log(sin3x)+c#

Hence we have that

#f(x)=-1/3cos(x^3)-1/3*log(sin3x)+c=>
f(pi/12)=-1/3cos(pi^3/12^3)-1/3*log(sin(3*pi/12))+c=>

Hence the function is

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To find the integral of ( f(x) = \int (x^2 \sin(x^3) - \cot(3x)) , dx ) given that ( f(\frac{\pi}{12}) = -1 ), we need to first integrate ( f(x) ) and then use the given value to determine the constant of integration.

After integrating ( f(x) ), we get:

[ F(x) = \frac{-\cos(x^3)}{3} - \frac{\ln|\sin(3x)|}{3} + C ]

Given that ( f(\frac{\pi}{12}) = -1 ), we can substitute ( x = \frac{\pi}{12} ) and ( F(\frac{\pi}{12}) = -1 ) into the integrated function and solve for the constant ( C ).

[ -\frac{\cos\left(\left(\frac{\pi}{12}\right)^3\right)}{3} - \frac{\ln|\sin\left(3\left(\frac{\pi}{12}\right)\right)|}{3} + C = -1 ]

Solve for ( C ) using the given equation. Then substitute ( C ) back into ( F(x) ) to get the final expression for ( f(x) ).

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To find (f(x)), integrate the given function (x^2\sin(x^3) - \cot(3x)) with respect to (x). Then, use the given condition (f(\frac{\pi}{12}) = -1) to solve for the constant of integration.

After integrating and solving for the constant of integration, we find that (f(x) = \frac{-\cos(x^3)}{3} - \frac{\ln|\sin(3x)|}{3} + C), where (C) is the constant of integration.

Given (f\left(\frac{\pi}{12}\right) = -1), substitute (x = \frac{\pi}{12}) into the expression for (f(x)) and solve for (C).

After substituting and solving, we find that (C = -\frac{2}{3}).

Therefore, the function (f(x) = \frac{-\cos(x^3)}{3} - \frac{\ln|\sin(3x)|}{3} - \frac{2}{3}).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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