Integrate 1/(1+x^2) with the substitution x=tan u

To solve the integral 

\[
\int \frac{1}{1+x^2} \, dx
\]

using the substitution \( x = \tan u \), we follow these steps:

1. **Substitution**:
   If \( x = \tan u \), then \( dx = \sec^2 u \, du \).

2. **Change the integral**:
   Substituting \( x \) and \( dx \) into the integral gives:

   \[
   \int \frac{1}{1+\tan^2 u} \sec^2 u \, du
   \]

3. **Simplify using the identity**:
   Recall the trigonometric identity \( 1 + \tan^2 u = \sec^2 u \). Thus, the integral simplifies to:

   \[
   \int \frac{\sec^2 u}{\sec^2 u} \, du = \int 1 \, du
   \]

4. **Integrate**:
   The integral of \( 1 \) with respect to \( u \) is:

   \[
   u + C
   \]

5. **Back-substitution**:
   Since \( u = \tan^{-1}(x) \), we have:

   \[
   \int \frac{1}{1+x^2} \, dx = \tan^{-1}(x) + C
   \]

Thus, the final answer is:

\[
\int \frac{1}{1+x^2} \, dx = \tan^{-1}(x) + C
\]

### Questions
1. Would you like to see a graphical representation of the integral?
2. Do you want to explore other substitution methods for this integral?
3. Would you like to see the application of this integral in real-world problems?
4. Do you want to learn about the properties of the arctangent function?
5. Would you like to practice similar integrals?

### Tip
When working with trigonometric substitutions, it's helpful to remember the identities associated with them, as they can greatly simplify the integration process.

Explore the integration of 1/(1+x^2) using the substitution x=tan u. This step-by-step guide demonstrates key calculus concepts, including trigonometric identities and the substitution method, suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation