Evaluate the following using The Substitution Rule and the Fundamental Theorem of Calculus.

Question

Evaluate the following using The Substitution Rule and the Fundamental Theorem of Calculus.

Problems is on this image

image- https://imgur.com/9qB5VSG

complete solution of 3 and 4

Answer 1

For problem 3 ∫_-ln3^ln3(e^x/e^x +4)dx becomes ∫_-ln3^ln3e^x/e^xdx + ∫_-ln3^ln34dx. This now becomes ln e^x|_-ln3^ln3 +8 ln3. The first term can be written as ln e^ln3 - e^-ln3 or ln3 -ln(1/3) using this gives ln3 -(ln1-ln3) +8ln3 which simplifies to 10 ln3 as the answer. For problem 4 the integral of the csc²(2x) is just - cot(2x)/2. So -(3/2)(cot(π/2) - cot(π/4)) gives -(3/2)(0-1) which becomes 3/2 as the answer.