Evaluate Integral

[danielatha4]

1. The problem statement, all variables and given/known data
Evaluate \int\frac{5x+5}{x^2+1}


2. Relevant equations



3. The attempt at a solution
5*\int\frac{x+1}{x^2+1}

5*\int\frac{x}{x^2+1}+\int\frac{1}{x^2+1}

The first term's value is (1/2)ln(x2+1) but what is the second term?

[iomtt6076]

for the second term, consult a table of integrals

[Mark44]

arctan(x)
Don't forget that both antiderivatives are multiplied by 5, and don't forget your constant of integration.

[danielatha4]

We were never instructed to refer to any tables, and I don't suspect that we should have to. And we haven't done anything as complex as arctan(x) yet.

The method to evaluate the integral should be fairly simple. It's the beginning of a calculus 2 class.

[snipez90]

Yes well that doesn't really change the fact that the antiderivative of 1/(1+x^2) is arctan(x) does it? And arctan(x) is not that complex, it's actually quite simple.

[danielatha4]

I'm not doubting that the antiderivative of 1/(x^2+1) is arctan(x). That's just not the method my teacher wants me to use because haven't learned inverse trig functions yet. Maybe I went about the problem the wrong way from the beginning?

[Dick]

I'm not doubting that the antiderivative of 1/(x^2+1) is arctan(x). That's just not the method my teacher wants me to use because haven't learned inverse trig functions yet. Maybe I went about the problem the wrong way from the beginning?

You did it exactly right. If you don't know the antiderivative is arctan(x) then you have to derive it using a trig substitution. Put x=tan(u).

[vela]

We were never instructed to refer to any tables, and I don't suspect that we should have to. And we haven't done anything as complex as arctan(x) yet.

The method to evaluate the integral should be fairly simple. It's the beginning of a calculus 2 class.
You probably did learn how to differentiate arctan(x) last semester. If you recognized the integrand was the derivative, you could just write the answer down for the second integral.

Have you learned using trig substitutions to do integrals yet?