Michael Redei said:No idea if this would help, but I'd try writing dt/du = 1/(1+u²) as dt = du/(1+u²) and substitute that in the integral ##\int_0^{\pi/2}\frac{-1}{1+x(\sin t)^2}dt##.
Out of curiosity, what is the answer?MareMaris said:In the end to get sin(t) and cos(t) in terms of u I just drew a right angle triangle with the side opposite to the hypotenuse as u and the side adjacent to it as 1, and worked it out using trig, seemed easier than using the identities and double angle rules that potentially you have to use because I know you use those to work out sin(t) and cos(t) with you use t sub.
I managed to get to an answer using sin(t) = u/(1+u^2)^1/2
SammyS said:Out of curiosity, what is the answer?
I get the same.MareMaris said:In the end after substituting everything in I got -pi/2root(1+x)
Integration by substitution is a technique used to simplify and solve integrals by substituting a new variable for the original variable in the integrand.
Using the substitution u=tan(t) can be helpful when the integrand contains trigonometric functions such as sine, cosine, or tangent. This substitution allows us to rewrite the integrand in terms of the new variable u, making it easier to integrate.
You can use integration by substitution whenever the integrand contains a function that can be expressed in terms of the new variable u. In the case of u=tan(t), this substitution is useful when the integrand contains trigonometric functions or expressions that can be rewritten in terms of tangent.
The steps for solving an integral using u=tan(t) substitution are as follows: 1. Identify the appropriate substitution u=tan(t).2. Rewrite the integrand in terms of u.3. Substitute u back into the integral.4. Integrate the new expression with respect to u.5. Rewrite the final answer in terms of the original variable.
Yes, there are some limitations to using u=tan(t) substitution. This substitution may not work for all integrals, especially if the integrand contains complicated expressions that cannot be rewritten in terms of tangent. In these cases, other substitution techniques or integration methods may be more appropriate.