# What is #f(x) = int 2sinx-xcosx dx# if #f((7pi)/6) = 0 #?

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The function ( f(x) = \int (2\sin(x) - x\cos(x)) , dx ) satisfies ( f\left(\frac{7\pi}{6}\right) = 0 ).

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To find ( f(x) = \int (2 \sin(x) - x \cos(x)) , dx ) when ( f\left(\frac{7\pi}{6}\right) = 0 ), first, find the antiderivative of ( 2\sin(x) - x\cos(x) ). Then use the given value to solve for the constant of integration. Finally, evaluate the definite integral at ( \frac{7\pi}{6} ).

The antiderivative of ( 2\sin(x) - x\cos(x) ) is ( -2\cos(x) - x\sin(x) + C ), where ( C ) is the constant of integration.

Since ( f\left(\frac{7\pi}{6}\right) = 0 ), we have: [ 0 = -2\cos\left(\frac{7\pi}{6}\right) - \frac{7\pi}{6}\sin\left(\frac{7\pi}{6}\right) + C ]

Now, evaluate the expression and solve for ( C ): [ 0 = -2\left(-\frac{\sqrt{3}}{2}\right) - \frac{7\pi}{6}\left(-\frac{1}{2}\right) + C ] [ 0 = \sqrt{3} + \frac{7\pi}{12} + C ] [ C = -\sqrt{3} - \frac{7\pi}{12} ]

Substitute ( C ) back into the antiderivative: [ f(x) = -2\cos(x) - x\sin(x) - \sqrt{3} - \frac{7\pi}{12} ]

Evaluate the definite integral at ( x = \frac{7\pi}{6} ): [ f\left(\frac{7\pi}{6}\right) = -2\cos\left(\frac{7\pi}{6}\right) - \frac{7\pi}{6}\sin\left(\frac{7\pi}{6}\right) - \sqrt{3} - \frac{7\pi}{12} ]

Since ( f\left(\frac{7\pi}{6}\right) = 0 ), we have: [ 0 = -2\left(-\frac{\sqrt{3}}{2}\right) - \frac{7\pi}{6}\left(-\frac{1}{2}\right) - \sqrt{3} - \frac{7\pi}{12} ] [ 0 = \sqrt{3} + \frac{7\pi}{12} - \sqrt{3} - \frac{7\pi}{12} ]

Hence, the value of ( f\left(\frac{7\pi}{6}\right) ) is indeed 0.

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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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