Integration using Trigonometric Substitution

When it comes to integrating functions that involve radicals or expressions like $a^2-x^2$, we often rely on trigonometric substitution to simplify the problem. Trigonometric substitution is a method that involves substituting a variable with a trigonometric function to make the integral easier to solve. In this article, we will explore how to use trigonometric substitution to evaluate integrals of the form:

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

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals by substituting the variable with a trigonometric function. In particular, when we have an integral of the form:

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

we can use the substitution $x = a\sin\theta$ or $x = a\cos\theta$ to simplify the problem. To determine which substitution to use, we need to look at the expression under the square root and compare it to the trigonometric identities:

$\sin^2\theta + \cos^2\theta = 1$

If we have an expression of the form $\sqrt{a^2-x^2}$, we should use the substitution $x = a\sin\theta$, which simplifies the expression to $\sqrt{a^2-a^2\sin^2\theta} = a\cos\theta$. On the other hand, if we have an expression of the form $\sqrt{x^2-a^2}$, we should use the substitution $x = a\cos\theta$, which simplifies the expression to $\sqrt{a^2\cos^2\theta-a^2} = a\sin\theta$.

Example

Let's apply these ideas to the integral:

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

Using the substitution $x = a\sin\theta$, we have:

$dx = a\cos\theta \ d\theta$

and

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

Substituting these expressions into the integral, we get:

$\int \sqrt{a^2-x^2} \ dx = \int a\cos^2\theta \ d\theta$

Now, we can use the identity $\cos^2\theta = \frac{1+\cos(2\theta)}{2}$ to simplify the integral further:

$\int a\cos^2\theta \ d\theta = \frac{a}{2}\int (1+\cos(2\theta)) \ d\theta$

Integrating each term separately, we get:

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

Substituting back for $x$, we get:

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

Conclusion

Trigonometric substitution is a powerful technique for evaluating integrals involving radicals or expressions like $a^2-x^2$. By substituting the variable with a trigonometric function, we can simplify the problem and use trigonometric identities to evaluate the integral. In particular, when we have an integral of the form $\int \sqrt{a^2-x^2} \ dx$, we can use the substitution $x = a\sin\theta$ to simplify the problem.

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