Dick said:Hey, hey. Sorry. I wasn't criticizing you. I can't TeX for s**t either. I'm just saying the problem is easier than you think. You just didn't put the integrand together with the dx and get a simple solution.
Trigonometric substitution is a technique used in calculus to simplify and solve integrals involving algebraic expressions and trigonometric functions.
Trigonometric substitution is useful when the integrand contains a square root of a quadratic expression, or when the integrand contains a sum or difference of two squares.
To perform a trigonometric substitution, you first identify which trigonometric function to substitute in for the variable. Then, you use a trigonometric identity to rewrite the integrand in terms of the new variable. Finally, you solve the integral using standard integration techniques.
The most common trigonometric substitutions are:
1. For expressions containing √(a²-x²): use x = a sinθ or x = a cosθ
2. For expressions containing √(a²+x²): use x = a tanθ or x = a cotθ
3. For expressions containing √(x²-a²): use x = a secθ or x = a cscθ
Some tips include:
1. Always choose the substitution that will make the integral simpler.
2. Use trigonometric identities to rewrite the integrand in terms of the new variable.
3. Don't forget to substitute back for the original variable at the end.
4. Practice using different substitutions to become more familiar with the technique.
5. Check your answer by differentiating it to ensure it is correct.