3.3 Q-13

Question Statement

Evaluate the following integral:

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

---

Background and Explanation

This problem involves the integration of a rational function containing a square root. To solve this, we will apply a substitution and trigonometric identities to simplify the expression into a more manageable form. The key steps involve recognizing the structure and transforming the integral using a substitution for easier integration.

---

Solution

Let’s solve the integral step by step:

1. Rewrite the Integral: The given integral is:

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

We notice that $x^4$ can be written as $(x^2)^2$. So, the integral becomes:

$$\int \frac{a x}{\sqrt{a^2 - (x^2)^2}} , dx$$

2. Substitute with Trigonometric Identity: To simplify, we use a trigonometric substitution. Let:

$$x^2 = a \sin \theta$$

which implies that:

$$2 x , dx = a \cos \theta , d\theta$$

3. Substitute into the Integral: Now, substitute into the integral. First, simplify the square root term:

$$\sqrt{a^2 - x^4} = \sqrt{a^2 - a^2 \sin^2 \theta} = a \cos \theta$$

Substituting into the integral:

$$\int \frac{a x}{a \cos \theta} \cdot \frac{a \cos \theta}{2} , d\theta$$

4. Simplify the Expression: After substitution, the integral simplifies to:

$$= \frac{a}{2} \int 1 , d\theta$$

The integral of 1 with respect to $\theta$ is simply $\theta$. Therefore:

$$= \frac{a}{2} \theta + C$$

5. Substitute Back $\theta$ in Terms of $x$: Recall that:

$$\sin \theta = \frac{x^2}{a}$$

So the solution becomes:

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

---

Key Formulas or Methods Used

• Substitution:
  • $x^2 = a \sin \theta$, which leads to $2 x , dx = a \cos \theta , d\theta$.
• Trigonometric Identity:
  • $\sqrt{a^2 - x^4} = a \cos \theta$.
• Inverse Trigonometric Function:
  • The integral of 1 with respect to $\theta$ is $\theta$, and we use $\sin \theta = \frac{x^2}{a}$ to find the solution in terms of $x$.

---

Summary of Steps

1. Rewrite the integral using $x^4 = (x^2)^2$.
2. Use the substitution $x^2 = a \sin \theta$, and simplify the square root term.
3. Substitute into the integral and simplify the expression.
4. Integrate and then substitute back $\sin \theta = \frac{x^2}{a}$ to express the result in terms of $x$.
5. The final result is:

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