# What is #f(x) = int xsin2x+cos3x dx# if #f((7pi)/12)=14 #?

We use the Rule of Integration by Parts (IBP) in the following

form:

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To find ( f(x) = \int xsin(2x) + cos(3x) , dx ) if ( f\left(\frac{7\pi}{12}\right) = 14 ), you can differentiate the function ( f(x) ) and solve for the constant of integration using the given information.

Differentiating ( f(x) ) will give you ( f'(x) = x \cdot \sin(2x) + \cos(3x) ).

Integrating ( f'(x) ) will give you ( f(x) = \int x \sin(2x) + \cos(3x) , dx ).

Then, evaluate ( f(x) ) at ( x = \frac{7\pi}{12} ) and set it equal to 14 to find the constant of integration.

Once you have the constant of integration, you can find the specific function ( f(x) ).

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