How do you evaluate the integral of #(x + (15/(sin^2(x))))# from pi/6 to pi/2?
We have
Which can be written as
Both of these terms can be evaluated separately (for now, indefinitely, we will evaluate the integral directly following this):
So, we want to evaluate
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To evaluate the integral of ( x + \frac{15}{{\sin^2(x)}} ) from ( \frac{\pi}{6} ) to ( \frac{\pi}{2} ), you would first find the antiderivative of the function, then evaluate it at the upper and lower limits of integration, and finally subtract the lower value from the upper value. This process is known as the definite integral.
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To evaluate the integral of ( x + \frac{15}{{\sin^2(x)}} ) from ( \frac{\pi}{6} ) to ( \frac{\pi}{2} ), you would need to integrate the function with respect to ( x ) over the given interval. Then, substitute the upper limit of integration (( \frac{\pi}{2} )) and subtract the result when you substitute the lower limit of integration (( \frac{\pi}{6} )). The integral of ( x ) with respect to ( x ) is ( \frac{x^2}{2} ). The integral of ( \frac{15}{{\sin^2(x)}} ) requires a trigonometric substitution. Let ( u = \sin(x) ), then ( du = \cos(x) , dx ). Using this substitution, you can rewrite the integral in terms of ( u ). Once integrated, you can revert to the variable ( x ) using the original substitution.
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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.
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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