Trigonometric substitution
To solve some problems in calculus, we can substitute the trigonometric functions. We can substitute the trigonometric function with the help of trigonometric identities. Those are,
Sin ^2 x + cos^2 x = 1, sec^2 x – tan^2 x = 1 and cosec^2 x – cot^2 x = 1. If square root of (a^2 – x^2) is there, we can substitute x = a sin y, then that will becomes a cos y. Then we can integrate and get the solution. If square root of (a^2 + x^2) is there, we can substitute x = a tan y, then that will becomes a sec y. Then we can integrate and get the solution. If square root of (x^2 – a^2) is there, we can substitute x = a sec y, then that will becomes a tan y. Then we can integrate and get the solution.
Examples:
1. Evaluate, Integration square root of (16 - x^2) dx.
2. Evaluate Integration (1 over square root of (a^2 + x^2)) dx.