3.3 Q-10

Question Statement

Evaluate the following integral:

$$\int \frac{1}{(1 + x^2) \tan^{-1} x} , dx$$

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Background and Explanation

This problem involves an integral that can be simplified using substitution. The key concept here is recognizing that the expression inside the integral involves an inverse trigonometric function, specifically $\tan^{-1} x$, and we can simplify it using an appropriate substitution. In this case, we can set $u = \tan^{-1} x$, which will transform the integral into a more manageable form.

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Solution

Let’s break down the solution step by step:

1. Substitute $u = \tan^{-1} x$: By letting $u = \tan^{-1} x$, we can rewrite the integral in terms of $u$. This substitution simplifies the inverse tangent term and allows us to proceed with a more straightforward integral.

$$u = \tan^{-1} x$$

2. Differentiate $u = \tan^{-1} x$ to find $du$: Taking the derivative of both sides with respect to $x$, we get:

$$du = \frac{1}{1 + x^2} , dx$$

3. Rewrite the integral: Now substitute $du = \frac{1}{1 + x^2} , dx$ into the original integral:

$$\int \frac{1}{(1 + x^2) \tan^{-1} x} , dx = \int \frac{1}{u} , du$$

4. Integrate: The integral of $\frac{1}{u}$ is a standard result:

$$\int \frac{1}{u} , du = \ln |u| + C$$

5. Substitute back $u = \tan^{-1} x$: Now, substitute $u = \tan^{-1} x$ back into the solution:

$$\ln |u| + C = \ln |\tan^{-1} x| + C$$

6. Final result: Therefore, the final result is:

$$\ln \left( \tan^{-1} x \right) + C$$

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Key Formulas or Methods Used

• Substitution method:
  • Let $u = \tan^{-1} x$, which simplifies the integral.
• Standard integral:
  • The integral of $\frac{1}{u}$ is $\ln |u| + C$.

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Summary of Steps

1. Substitute $u = \tan^{-1} x$.
2. Differentiate to find $du = \frac{1}{1 + x^2} , dx$.
3. Rewrite the integral as $\int \frac{1}{u} , du$.
4. Integrate to get $\ln |u| + C$.
5. Substitute back $u = \tan^{-1} x$.
6. Final result: $\ln \left( \tan^{-1} x \right) + C$.