When should you use trig substitution?

As we saw in class, you can use trig substitution even when you don’t have square roots. In particular, if you have an integrand that looks like an expression inside the square roots shown in the above table, then you can use trig substitution. You should only do so if no other technique (e.g., u-substitution) works.

What is trig substitution in calculus?

In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions.

How does trigonometric substitution work?

When x2+a2 is embedded in the integrand, use x=atan(θ). To convert back to x, use your substitution to get xa=tanθ, and draw a right triangle with opposite side x, adjacent side a and hypotenuse √x2+a2. When a2−x2 is embedded in the integrand, use x=asin(θ).

Why do we use trig substitution and why do we use the right triangle?

It’s like u-substitution, integration by parts, or partial fractions. It takes advantage of the relationship between the sides and angles in a right triangle, allowing you to replace a more complicated value in an integral, with simpler associated values from a corresponding right triangle.

Why do we use trigonometric substitution in integration?

What is the goal of trigonometric substitution Why does this work for integration?

These fancy functions involve things like a2 + x2 or a2 – x2 or x2 – a2 , usually under root signs or inside half-powers, and the purpose of trig substitution is to use the magic of trig identities to make the roots and half-powers go away, thus making the integral easier.

Why Does trig substitution work for integration?

This works because the unit circle has two different kinds of parametrizations: one algebraic and the other trigonometric. You can write all points on the unit circle in the form (cos(𝜃), sin(𝜃)) as 𝜃 runs over the real numbers.