jtt said:Homework Statement
use substitution to evaluate the integralHomework Equations
1)∫ tan(4x+2)dx
2)∫3(sin x)^-2 dx
The Attempt at a Solution
1) u= 4x+2 du= 4
(1/4)∫4 tan(4x+2) dx
∫(1/4)tan(4x+2)(4dx)
∫ (1/4) tanu du
(1/4)ln ltan(u)l +c
2) u=sinx du= cosx or u=x du = 1 ?
Homework Statement
Homework Equations
The Attempt at a Solution
The general rule for using u-substitution with trig functions is when you have an integral that involves a trig function and its derivative. In other words, if the function inside the integral sign is a trig function and the function's derivative is also present in the integral, you can use u-substitution to simplify the integral.
The substitution variable (u) should be chosen so that when you take the derivative of u, it will cancel out with the rest of the integral. In general, a good choice for u is a part of the integral that is not already present or can be easily manipulated to match the derivative.
Yes, u-substitution with trig functions can also be used for definite integrals. The limits of integration will need to be changed to match the substitution variable (u).
Yes, there are a few special cases when using u-substitution with trig functions. One is when the integral involves both a trig function and its inverse, in which case a trig identity may need to be used. Another case is when the integral involves a trig function raised to an odd power, in which case the substitution may need to be modified.
To check if your u-substitution with trig is correct, simply take the derivative of u and see if it matches the function inside the integral. If it does, then the substitution was successful. You can also plug in the original function and limits of integration into the integral with the substituted variable u and see if it matches the original integral.