Example Trigonometric Substitutions

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Example Trigonometric Substitutions

In Calculus, an Example Trigonometric Substitutions; Tutoring Math such as calculus, the need for examples to convey the procedure is essential for students understanding of the process. Such as Trigonometric substitution; this procedure is for simplifying complex integrations that make them easier to evaluate. See the following blog about the process for Trigonometric Substitution. The example provided seems simple integration but at a closer look, the function doesn’t fit into a standard format.

Let’s look at the following problem:

\int\sqrt{a^2-u^2} du

 {u} = {a}\sin(\Theta) or \frac{u}{a} = \sin(\Theta) and \sqrt{a^2-u^2} = {a}\cos(\Theta) ; \large \cos\Theta = \frac{\sqrt{a^2-u^2}}{a} we can check this result by Trionometric Substituting of {a}\sin(\Theta) into the express for the hypotenuse \sqrt{a^2-(\\(a)^2(sin(\Theta))^2} = {a}\sqrt{1-(\sin(\Theta))^2} = {a}\cos(\Theta) from the indemnity above {du} = {a}\cos(\Theta)d(\Theta)

Therefore; \int\sqrt{a^2-u^2}du = \int{a}cos(\Theta){a}cos(\Theta)d(\Theta) = \int{a^2}cos^2(\Theta)d(\Theta) we can use the double angle  cos({2\Theta}) = \frac{1 + cos({2\Theta})}{2} formula for simplify the expression before we integrate and applying the identity for double angle for \sin({2\Theta}) = \sin(\Theta)cos(\Theta) ;

Therefore;

{a}^2\int\frac{(1 + cos(2\Theta))}{2}d(\Theta) = \frac{a^2}{2}((\Theta)+ \frac{1}{2}sin(2\Theta))+ C = \frac{a^2}{2}((\Theta)+\frac{1}{2}sin(\Theta)cos(\Theta)) + C

Also, remember that with definite integrals the limits of integration =need to be changed from u to θ.
For this example, \Theta = \arcsin \frac{u}{a} and \sin(\Theta) are bounded between -π/2 ≤ θ ≤π/2.

Therefore, this completes this Example of Trigonometric Substitution and reveals this techniques process and power for these types of problems.

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See you next time.

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