3.3 Q-1

Question Statement

Evaluate the integral:

\int \frac{-2x}{\sqrt{4 - x^2}} , dx

Background and Explanation

In this problem, we are dealing with an integral that involves a square root in the denominator and a linear term in the numerator. To solve this, we’ll use substitution and standard integration techniques.

To tackle integrals like these, it is often helpful to recognize a pattern or to simplify the expression using trigonometric substitution or algebraic manipulation. In this case, the square root suggests the use of a standard formula for integrals of the form $\int \frac{x}{\sqrt{a^2 - x^2}} , dx$ .

Solution

Rewrite the integral:
The given integral is:

\int \frac{-2x}{\sqrt{4 - x^2}} , dx

This matches the form of the standard integral, so we proceed with substitution.

Substitute to simplify:
Notice that the integral has a structure that hints at a simple substitution. We recognize that the expression under the square root is of the form $4 - x^2$ . This suggests the substitution:

u = 4 - x^2 \quad \text{so that} \quad du = -2x , dx

The differential $-2x , dx$ matches the numerator of the integral, so we can directly substitute:

\int \frac{-2x}{\sqrt{4 - x^2}} , dx = \int \frac{\sqrt{u}}{\sqrt{u}} , du = \int u^{1/2} , du

Integrate:
Now, integrate the simplified expression:

\int u^{1/2} , du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2}

Substitute back:
Finally, substitute back the value of $u$ (which is $4 - x^2$ ) to return to the variable $x$ :

\frac{2}{3} (4 - x^2)^{3/2} + C

Therefore, the final answer is:

2 \sqrt{4 - x^2} + C

Key Formulas or Methods Used

Substitution: We used the substitution $u = 4 - x^2$ to simplify the integral.
Power Rule: The integral of $u^{1/2}$ was computed using the power rule for integrals.

Summary of Steps

Recognize the form of the integral and apply the substitution $u = 4 - x^2$ .
Simplify the integral and integrate $u^{1/2}$ .
Substitute back to express the result in terms of $x$ .
Final answer: $2 \sqrt{4 - x^2} + C$ .