Help in Calculus Problem

Question

I've been trying this problem for a long time and can't answer it. I know it has to do with trig functions and u-substitutions but that's all I know.

Answer 1

The analytic approach to this integral starts with the u substitution u = 2.343458x-.812495

The indefinite form then involves ∫ sqrt(1 + u²) du / 2.343458

The anti-derivative of sqrt( 1 + u²) is (1/2) x sqrt(1 + x²) + (1/2)ln(u + sqrt(1 + u²) ) +C

or (1/2) x sqrt(1 + u²) + sinh^-1(x)

As you can see from this form, it is derivable by methods of trig substitution, but is pretty complicated.

Since the required integral is a definite integral, it is easier to get an answer using a TI-84 calculator

I got 1.071750442

Answer 2

I'm not sure I have all of this for you, but here are some hints.

First make the substitution u = ax - b where a and b are those 2 horrendous constants.

That makes the integrand (1/a) sqrt(1+u²) du.

Then u = tan θ makes the integrand (1/a) sec³θ dθ which can be integrated by a reduction formula (look it up!!!).

That gives you the integral, now just work backwards and use those awful constants and integration limits.

Maybe there is some easy trick that has to do with the constants but I don't see it.

Good luck!