How do you use the Fundamental Theorem of Calculus to find the derivative of #int xsin(πx) # from 1 to x^2?
Strictly speaking you don't, because you can't have the variable you're integrating show up in the limits of integration. Assuming you meant something like
We see that the fundamental theorem of calculus states that
Now, we just differentiate both sides
Using the chain rule:
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To use the Fundamental Theorem of Calculus to find the derivative of ( \int_{1}^{x^2} x \sin(\pi x) , dx ), follow these steps:

Evaluate the antiderivative of the integrand ( x \sin(\pi x) ): [ \int x \sin(\pi x) , dx = \frac{x \cos(\pi x)}{\pi} + \frac{\sin(\pi x)}{\pi} + C ]

Substitute the upper limit (( x^2 )) into the antiderivative expression: [ \frac{x^2 \cos(\pi x^2)}{\pi} + \frac{\sin(\pi x^2)}{\pi} + C ]

Substitute the lower limit (( 1 )) into the antiderivative expression: [ \frac{1 \cdot \cos(\pi)}{\pi} + \frac{\sin(\pi)}{\pi} + C ]

Simplify: [ \frac{\cos(\pi)}{\pi} + \frac{\sin(\pi)}{\pi} + C ]

Take the derivative of the expression obtained in steps 2 to 4 with respect to ( x ): [ \frac{d}{dx} \left( \frac{x^2 \cos(\pi x^2)}{\pi} + \frac{\sin(\pi x^2)}{\pi} + C \right) ]

Simplify the derivative: [ \frac{2x \cos(\pi x^2)}{\pi}  2x \sin(\pi x^2) ]
Therefore, the derivative of ( \int_{1}^{x^2} x \sin(\pi x) , dx ) is ( \frac{2x \cos(\pi x^2)}{\pi}  2x \sin(\pi x^2) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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