3.4 Q-4

Question Statement

Evaluate the following integrals:

i. $\int \sqrt{a^{2}-x^{2}} , dx$

ii. $\int \sqrt{x^{2}-a^{2}} , dx$

iii. $\int \sqrt{4-5 x^{2}} , dx$

iv. $\int \sqrt{3-4 x^{2}} , dx$

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Background and Explanation

To solve these integrals, we need to use various techniques from calculus, particularly integration methods involving square roots of quadratic expressions. These integrals are typically solved using trigonometric substitution or by recognizing them as standard forms.

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Solution

i. $\int \sqrt{a^{2}-x^{2}} , dx$

We start by applying integration by parts and simplifying the expression step-by-step:

1. Let $I = \int \sqrt{a^2 - x^2} , dx$
2. Use integration by parts: $I = x \sqrt{a^2 - x^2} - \int \frac{x^2}{\sqrt{a^2 - x^2}} , dx$
3. Simplify the remaining integral by rewriting it: $I = x \sqrt{a^2 - x^2} + \int \frac{a^2 - (a^2 - x^2)}{\sqrt{a^2 - x^2}} , dx$
4. This leads to: $I = x \sqrt{a^2 - x^2} + \int \frac{a^2}{\sqrt{a^2 - x^2}} , dx - I$
5. Solve for $I$: $2I = x \sqrt{a^2 - x^2} + a^2 \sin^{-1}\left(\frac{x}{a}\right) + C_1$
6. Thus, the solution is: $I = \frac{1}{2} \left[x \sqrt{a^2 - x^2} + a^2 \sin^{-1}\left(\frac{x}{a}\right)\right] + C$

ii. $\int \sqrt{x^{2}-a^{2}} , dx$

For this integral, we follow a similar approach with trigonometric substitution:

1. Let $I = \int \sqrt{x^2 - a^2} , dx$
2. Apply integration by parts: $I = x \sqrt{x^2 - a^2} - \int \frac{x^2}{\sqrt{x^2 - a^2}} , dx$
3. Simplify the remaining integral: $I = x \sqrt{x^2 - a^2} - \int \frac{x^2 - a^2 + a^2}{\sqrt{x^2 - a^2}} , dx$
4. Break it into simpler parts: $I = x \sqrt{x^2 - a^2} - I - a^2 \int \frac{1}{\sqrt{x^2 - a^2}} , dx$
5. Solve for $I$: $2I = x \sqrt{x^2 - a^2} - a^2 \ln \left| x + \sqrt{x^2 - a^2} \right| + C_2$
6. Thus, the solution is: $I = \frac{1}{2} \left[x \sqrt{x^2 - a^2} - a^2 \ln \left| x + \sqrt{x^2 - a^2} \right|\right] + C$

iii. $\int \sqrt{4-5 x^{2}} , dx$

In this case, we use substitution to simplify the integral:

1. Let $I = \sqrt{5} \int \sqrt{\left(\frac{4}{5} - x^2\right)} , dx$
2. Perform trigonometric substitution: $I = \sqrt{5} \left[\frac{x}{2} \sqrt{\left(\frac{4}{5} - x^2\right)} + \frac{1}{2} \left(\frac{2}{\sqrt{5}}\right)^2 \sin^{-1}\left(\frac{x}{\frac{2}{\sqrt{5}}}\right)\right] + C$
3. Simplify further: $I = \frac{x}{2} \sqrt{4 - 5x^2} + \frac{2}{\sqrt{5}} \sin^{-1}\left(\frac{\sqrt{5}x}{2}\right) + C$

iv. $\int \sqrt{3-4 x^{2}} , dx$

For this integral, we apply a substitution and integration by parts:

1. Let $I = \int \sqrt{3 - 4x^2} , dx$
2. Use substitution with $y = 2x$, leading to: $I = \frac{1}{2} \left[ x \sqrt{3 - 4x^2} + \frac{3}{4} \sin^{-1}\left(\frac{2x}{\sqrt{3}}\right)\right] + C$

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Key Formulas or Methods Used

1. Integration by Parts: $\int u , dv = uv - \int v , du$

2. Trigonometric Substitution: For integrals of the form $\int \sqrt{a^2 - x^2} , dx$, use $x = a \sin \theta$ or similar substitutions.

3. Inverse Trigonometric Function: $\sin^{-1} \left( \frac{x}{a} \right)$

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Summary of Steps

1. For i:

   • Apply integration by parts.
   • Simplify using standard forms.
   • Solve for $I$ to get the final expression.

2. For ii:

   • Similar process as in part (i), but with different functions.
   • Use integration by parts and solve for $I$.

3. For iii:

   • Apply substitution to simplify the integral.
   • Use inverse trigonometric functions to finish the solution.

4. For iv:

   • Use substitution and apply integration by parts.
   • Simplify the expression using standard methods.

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Reference

By Great Science Academy: