danielatha4 said: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 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.danielatha4 said: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.
The integral of 5*ln(x^2+1) is equal to x^2*ln(x^2+1) + 2x + C.
To evaluate this integral, you can use the substitution method. Let u = x^2+1, then du/dx = 2x and dx = du/2x. Substituting these values into the integral, we get 5*ln(u)*(du/2x). Simplifying further, we get 5*ln(u)/2 + C = 5*ln(x^2+1)/2 + C.
Yes, the integral 5*ln(x^2+1) can be solved using integration by parts. Let u = ln(x^2+1), then du/dx = 2x/(x^2+1) and v = 5x, then dv/dx = 5. Substituting these values into the integration by parts formula, we get the final result of x^2*ln(x^2+1) + 2x + C.
The antiderivative of 5*ln(x^2+1) is equal to x^2*ln(x^2+1) + 2x + C.
Yes, the integral 5*ln(x^2+1) can be evaluated using a combination of substitution and integration by parts. This method is known as the double-substitution method and involves using substitution first, followed by another substitution or integration by parts to simplify the integral further.