Trigonometric Substitution

What is Trigonometric Substitution?

Trigonometric substitution is technique of integration used to solve the integrals involving  expressions with radicals and square roots such as √(x² + a²), √(a² + x²), and √(x² – a²).

When should I use Trigonometric Substitution?

Trigonometric substitution is useful when you have an integral that involves a radical expression, especially when the radical expression contains a quadratic term.

What are the Three Trigonometric Substitutions commonly used in Integrals?

The three commonly used trigonometric substitutions are:

• Substitute x = a sin θ when the radical expression contains a term of the form a² – x².
• Substitute x = a tan θ when the radical expression contains a term of the form x² – a².
• Substitute x = a sec θ when the radical expression contains a term of the form x² + a².

How does anyone choose which Trigonometric Substitution to Use?

You should choose the trigonometric substitution based on the form of the radical expression. If the radical expression contains a term of the form a^2 – x^2, use x = a sin θ. If the radical expression contains a term of the form x^2 – a^2, use x = a tan θ. If the radical expression contains a term of the form x^2 + a^2, use x = a sec θ.

Trigonometric Substitution is one of the substitution methods of integration where a function or expression in the given integral is substituted with trigonometric functions such as sin, cos, tan, etc. Integration by substitution is an easiest substitution method.

It is used when we make a substitution of a function, whose derivative is already included in the given integral function. By this, the function gets simplified, and simple integrals function is obtained which we can integrate easily. It is also known as u-substitution or the reverse chain rule. Or in other words, using this method, we can easily evaluate integrals and antiderivatives.