Determine the integral using trig substitution.
∫1/1−x2−−−−−√dx
We should use the trig substitution x=
sin(θ)
.
Hence dx=
cos(θ)
dθ
.

Expressing our integral in terms of θ
 gives:

∫1/1−x2−−−−−√dx=∫
dθ
and the antiderivative in terms of θ
 is
∫dθ=


We are tasked with evaluating the integral \( \int \frac{1}{\sqrt{1 - x^2}} \, dx \) using trigonometric substitution, where \( x = \sin(\theta) \). Let's go through the process step by step.

### Step 1: Substitution

Given that \( x = \sin(\theta) \), we have:

\[
dx = \cos(\theta) \, d\theta
\]

Now, substitute this into the integral:

\[
\sqrt{1 - x^2} = \sqrt{1 - \sin^2(\theta)} = \sqrt{\cos^2(\theta)} = \cos(\theta)
\]

### Step 2: Rewrite the Integral

Substituting into the integral:

\[
\int \frac{1}{\sqrt{1 - x^2}} \, dx = \int \frac{1}{\cos(\theta)} \cdot \cos(\theta) \, d\theta
\]

This simplifies to:

\[
\int d\theta
\]

### Step 3: Solve the Integral

The integral \( \int d\theta \) is straightforward:

\[
\int d\theta = \theta + C
\]

### Step 4: Back-Substitute for \( \theta \)

Since \( x = \sin(\theta) \), we have \( \theta = \arcsin(x) \). Therefore, the solution to the integral is:

\[
\theta + C = \arcsin(x) + C
\]

### Final Answer:

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

Would you like more details on any part of this solution? Here are some related questions for further exploration:

1. What other trigonometric identities can simplify integrals?
2. How do you use substitution in integrals involving \( \sec(\theta) \)?
3. How do you handle integrals of the form \( \int \frac{1}{\sqrt{a^2 - x^2}} \, dx \)?
4. What are some common applications of the \( \arcsin(x) \) function in physics?
5. What is the geometric interpretation of \( \arcsin(x) \) in terms of a right triangle?

**Tip:** Always check if the substitution simplifies both the numerator and denominator before proceeding!

Determine the integral using trig substitution. ∫1/√(1−x²) dx. We should use the trig substitution x = sin(θ). Hence, dx = cos(θ) dθ. Expressing our integral in terms of θ gives: ∫1/√(1−x²) dx = ∫ dθ and the antiderivative in terms of θ is ∫dθ = 

Explore how to evaluate the integral ∫1/√(1−x²) dx using trigonometric substitution with x = sin(θ). This problem includes step-by-step instructions, important formulas, and a breakdown of key trigonometric concepts. Ideal for students in Grades 11-12 and college-level calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation