How do you find the integral of #int 17/(16+x^2)dx#?

Answer 1

#17/4 arctan(x/4)+C#. Standard form: #int1/(a^2+x^2)dx=(1/a)arctan (x/a)+C#

Alternatively substitute #x=4tan u# and use #1+tan^2u=sec^2u# giving the integral as #int (17 xx 4)sec^2 u/(16 sec^2 u)du# and hence the answer.
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Answer 2

To find the integral of ( \int \frac{17}{16+x^2} , dx ):

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

  2. 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.

  3. 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 ).

  4. 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) ).
  5. 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 ).

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

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

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

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

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

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Answer from HIX Tutor

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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