Calculus Integration Trigonometric Substitution Techniques

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Calculus Integration Trigonometric Substitution Techniques is necessary for integrating functions such as the one provided below. This Substitution simplifies the integration by transforming a complex function into a trigonometric one. The use of a unit triangle as shown below makes the easily understood.

\int\frac{1}{\sqrt{(a^2 - u^2)}}du

Calculus Integration Trigonometric Substitution Techniques are more complex integration method than the first method that is learned of u-substitution but this doesn’t work for every situation. If you have something looks like a Pythagorean theorem is in use so that we can derive and easier solution method for ourselves.

Unit Triangle

Trigonometric substitution is one of the many techniques covered in the integral calculus. It involves the integrals containing; \sqrt{(a^2 - u^2)}, \sqrt{(a^2 + u^2)} , \sqrt{(u^2 - a^2)}

“u” is the variable and ”a” is a positive constant, and identities 1-sin(2θ) = cos(2θ) and 1-tan(2θ) = sec(2θ).
For occurrences of;
\sqrt{(a^2 - u^2)}}; u = a sin (θ) ; –π/2 ≤ θ ≤ π/2,
\sqrt{(a^2 + u^2)}}; u= a tan (θ) ; –π/2 ≤ θ ≤ π/2,
\sqrt{(u^2 - a^2)}}; u= a sec (θ); 0 ≤ θ ≤ π/2 if u ≥ a and π/2 ≤ θ ≤ π/2 if u ≤ -a

These problems can be very time consuming and tricky but if you follow some rules and slow down you will get through them and be able to handle anything your instructor throws at you.
The first step is the see if the problem is in a standard format such as the integral of \arcsin (\frac{u}{a}) , \frac{1}{a}\arctan (\frac{u}{a}) , or \frac{1}{a}{arcsec}\frac{u}{a} these are the results of integrals of\int\frac{1}{\sqrt{(a^2 - u^2)}}du , \int\frac{1}{\sqrt{(a^2 + u^2)}}du , \int\frac{1}{\sqrt{(u^2 - a^2)}}du respectively. If you these forms can’t be obtained by simple manipulation of the equations then the identities above should see which fits the situation the best.

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