How do you find the integral of #int 17/(16+x^2)dx#?
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To find the integral of ( \int \frac{17}{16+x^2} , dx ):

Recognize that the integrand resembles the form of the arctangent function.

Apply the substitution ( x = 4\tan(\theta) ) or ( x = 4\cot(\theta) ), which simplifies the integral into a form where you can use trigonometric identities.

After substitution, the integral becomes ( \int \frac{17}{16+16\tan^2(\theta)} \cdot 4,d\theta ) or ( \int \frac{17}{16+16\cot^2(\theta)} \cdot (4),d\theta ).

Simplify the integrand using trigonometric identities:
 For the ( \tan(\theta) ) substitution: ( 16+16\tan^2(\theta) = 16(1+\tan^2(\theta)) = 16\sec^2(\theta) ).
 For the ( \cot(\theta) ) substitution: ( 16+16\cot^2(\theta) = 16(1+\cot^2(\theta)) = 16\csc^2(\theta) ).

Rewrite the integral as ( \int \frac{17}{16\sec^2(\theta)} \cdot 4,d\theta ) or ( \int \frac{17}{16\csc^2(\theta)} \cdot (4),d\theta ).

Simplify the expression further to ( \int \frac{17\cos^2(\theta)}{4},d\theta ) or ( \int \frac{17\sin^2(\theta)}{4},d\theta ).

Integrate ( \frac{17\cos^2(\theta)}{4} ) or ( \frac{17\sin^2(\theta)}{4} ) with respect to ( \theta ).

Reverse the substitution to express the result in terms of ( x ).

The final answer is ( \frac{17}{4}(\theta + C) ), where ( C ) is the constant of integration.

If necessary, express the answer in terms of ( x ) using the original substitution.
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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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