3.3 Q-14

Question Statement

Evaluate the following integral:

$$\int \frac{dx}{\sqrt{7 - 6x - x^2}}$$

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Background and Explanation

This problem involves solving an integral with a square root of a quadratic expression. The method to simplify this is by completing the square for the quadratic expression under the square root, which allows us to apply a standard trigonometric substitution to evaluate the integral.

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Solution

Let’s solve the integral step by step:

1. Rewrite the Integral: The given integral is:

$$\int \frac{dx}{\sqrt{7 - 6x - x^2}}$$

First, we want to complete the square in the quadratic expression under the square root:

$$7 - 6x - x^2 = - (x^2 + 6x - 7)$$

2. Complete the Square: We complete the square for the expression $x^2 + 6x - 7$. To do this, we take half of the coefficient of $x$, square it, and add it and subtract it inside the expression:

$$x^2 + 6x = (x + 3)^2 - 9$$

Now substitute this back into the expression:

$$- (x^2 + 6x - 7) = - \left( (x + 3)^2 - 9 - 7 \right) = - \left( (x + 3)^2 - 16 \right)$$

So the integral becomes:

$$\int \frac{dx}{\sqrt{-(x + 3)^2 + 16}} = \int \frac{dx}{\sqrt{16 - (x + 3)^2}}$$

3. Apply the Trigonometric Substitution: We recognize this as a standard form that can be solved using a trigonometric substitution. Let:

$$x + 3 = 4 \sin \theta$$

This implies:

$$dx = 4 \cos \theta , d\theta$$

Substituting these into the integral:

$$\int \frac{4 \cos \theta , d\theta}{\sqrt{16 - 16 \sin^2 \theta}} = \int \frac{4 \cos \theta , d\theta}{4 \cos \theta} = \int d\theta$$

4. Integrate: The integral of $d\theta$ is simply $\theta$. Therefore:

$$\theta + C$$

5. Substitute Back in Terms of $x$: Recall the substitution $x + 3 = 4 \sin \theta$, so:

$$\sin \theta = \frac{x + 3}{4}$$

Therefore, the final result is:

$$\sin^{-1} \left( \frac{x + 3}{4} \right) + C$$

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Key Formulas or Methods Used

• Completing the Square:
  • To rewrite the quadratic expression as $(x + 3)^2 - 16$.
• Trigonometric Substitution:
  • Substituting $x + 3 = 4 \sin \theta$, which simplifies the integral.
• Inverse Sine:
  • The result of the integration leads to an inverse sine function.

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Summary of Steps

1. Rewrite the quadratic expression under the square root and complete the square.
2. Use the trigonometric substitution $x + 3 = 4 \sin \theta$.
3. Simplify the integral using the substitution.
4. Integrate, which gives $\theta + C$.
5. Substitute back in terms of $x$ to get the final result:

$$\sin^{-1} \left( \frac{x + 3}{4} \right) + C$$