3.3 Q-8

Question Statement

a. Show that:

$$\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln \left| \frac{x + \sqrt{x^2 - a^2}}{a} \right| + C$$

b. Show that:

$$\int \sqrt{a^2 - x^2} , dx = \frac{a^2}{2} \sin^{-1} \left( \frac{x}{a} \right) + \frac{x}{2} \sqrt{a^2 - x^2} + C$$

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Background and Explanation

To solve these integrals, we apply standard substitution techniques and trigonometric identities. The first integral involves the inverse hyperbolic trigonometric function, while the second uses trigonometric substitution. Both methods simplify the integrals into forms that are easier to integrate.

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Solution

Part a:

1. Substitution:
   We begin by making the substitution $x = a \sec \theta$. This leads to the following:

   • $dx = a \sec \theta \tan \theta , d\theta$
   • The integral becomes:

$$\int \frac{dx}{\sqrt{x^2 - a^2}} = \int \frac{a \sec \theta \tan \theta , d\theta}{\sqrt{(a \sec \theta)^2 - a^2}}$$

2. Simplification:
   Simplify the denominator:

$$\sqrt{(a \sec \theta)^2 - a^2} = \sqrt{a^2(\sec^2 \theta - 1)} = a \sqrt{\tan^2 \theta} = a \tan \theta$$

Thus, the integral becomes:

$$\int \frac{a \sec \theta \tan \theta , d\theta}{a \tan \theta} = \int \sec \theta , d\theta$$

3. Integrating:
   The integral of $\sec \theta$ is:

$$\int \sec \theta , d\theta = \ln |\sec \theta + \tan \theta| + C$$

4. Substitute back:
   Recall that $x = a \sec \theta$, so $\sec \theta = \frac{x}{a}$. Also, $\tan \theta = \sqrt{\sec^2 \theta - 1} = \sqrt{\frac{x^2}{a^2} - 1} = \frac{\sqrt{x^2 - a^2}}{a}$. Thus, the result is:

$$\ln \left| \frac{x + \sqrt{x^2 - a^2}}{a} \right| + C$$

Part b:

1. Substitution:
   Use the substitution $x = a \sin \theta$, so $dx = a \cos \theta , d\theta$. The integral becomes:

$$\int \sqrt{a^2 - x^2} , dx = \int \sqrt{a^2 - (a \sin \theta)^2} \cdot a \cos \theta , d\theta$$

2. Simplify:
   Simplifying the expression under the square root:

$$\sqrt{a^2 - a^2 \sin^2 \theta} = \sqrt{a^2 \cos^2 \theta} = a \cos \theta$$

Thus, the integral becomes:

$$a^2 \int \cos^2 \theta , d\theta$$

3. Use of identity:
   Apply the double-angle identity for cosine:

$$\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$$

The integral becomes:

$$a^2 \int \frac{1 + \cos 2\theta}{2} , d\theta$$

4. Integrating:
   Integrate term by term:

$$\int \frac{1 + \cos 2\theta}{2} , d\theta = \frac{\theta}{2} + \frac{\sin 2\theta}{4}$$

Therefore, the integral is:

$$\frac{a^2}{2} \theta + \frac{a^2}{4} \sin 2\theta + C$$

5. Substitute back:
   Now, use the inverse sine function for $\theta$, where $\sin \theta = \frac{x}{a}$. Also, use the identity $\sin 2\theta = 2 \sin \theta \cos \theta$. Finally, substitute back to express the result in terms of $x$:

$$\frac{a^2}{2} \sin^{-1} \left( \frac{x}{a} \right) + \frac{x}{2} \sqrt{a^2 - x^2} + C$$

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Key Formulas or Methods Used

• Substitution:
  We used trigonometric substitutions to simplify the integrals.

• Double-Angle Identity:
  The identity $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$ was applied to simplify the integral.

• Integration of Trigonometric Functions:
  The standard integrals of $\sec \theta$ and $\cos^2 \theta$ were used.

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Summary of Steps

Part a:

1. Use the substitution $x = a \sec \theta$.
2. Simplify the integral using trigonometric identities.
3. Integrate $\sec \theta$.
4. Substitute back in terms of $x$ to obtain the result.

Part b:

1. Use the substitution $x = a \sin \theta$.
2. Simplify the integral using trigonometric identities.
3. Integrate the expression.
4. Substitute back in terms of $x$ to obtain the final result.