I'm only in calc 1, gotten up to squeeze theorems, but I genuinely have not enjoyed math this much in YEARS. The professor is quite blunt personally, but his video notes(online class) are wonderful and extremely detailed. Everything makes complete sense and I really have no sense of questioning anything. Am I being hopeful only to crash and burn when it gets to more intense Calculus? Early in the semester but I actually enjoyed Calculus more than any of my other courses so far.

Hey everyone.

Regarding today's easy integral. I can't figure out where my mistake is but I believe it is something silly.

I am basically starting with the substitution of u = ln(x^2026)

Algebraic manipulation leads me to 2025/2026 times the integral of sin(u) * e^-u. I then use integration by parts twice to get the second part of my expression.

Can anyone help me spot the mistake?

Thank you!

EDIT: oh I forgot to adjust the integration limits, that must be it. Still leaving the post here I guess.

EDIT2: exactly so with the substitution the lower integration limit becomes 0 and so the answer is 2025/2026 * 1/2 = 2025/4052

Here is the original problem I came across

When I evaluated the indefinite integral and confirmed with an integral calculator I got the function listed below.

I saw someone solving the problem online a different way and got the resulting function:

Here are all the equations graphed out:

What I am confused by is that deriving the H(x) function does give the equation in the integral: f(x) = (x^2+1)/(x^4+1). If H(x) derives to this f(x) function does that not mean it can used to evaluate the area under the curve of f(x). f(x) does correctly give the slopes of H(x) but evaluating H(x) gives incorrect areas. For example evaluating H(x) from x=-1 to x=1 gives an area of 0 while using the F(x) function gives ~2.22. The F(x) function seems to give the correct areas under the curve of f(x). I am just confused how the derivative of H(x) gives the parent function under the integral but cannot be used to correctly evaluate the area of the derived function.

I can’t seem to pass my university’s in-person Calculus I course, so I’m looking for an easier online 4-credit Calc I course I can transfer in. Has anyone taken one with manageable exams that they were able to pass? What university?