Evalute the integral of x/(x^2+4)dx using u-substitution and then trigonometric substitution. I did this and got (1/2)ln(x^2+4)+C using u-substitution and x^2+4+C using trigonometric substitution. What did I do wrong??? thank you for helping!
The answer is clearly
(1/2) ln(x^2 + 4) + C
you can check by taking the derivative
I don't know how you got the other answer, but it is definitely incorrect
Let's go through both solutions step by step to identify any mistakes.
Using u-substitution:
1. Let's start by substituting u for x^2+4. Therefore, u = x^2+4.
2. Now, we need to find du/dx and solve for dx to rewrite the integral in terms of u. Taking the derivative of both sides, we get: du/dx = 2x.
Rearranging the equation, dx = du/(2x).
3. Substitute u and dx in the integral to get: ∫(x/(x^2+4))dx = ∫((1/2x)(du)).
4. Simplify the integral to: (1/2)∫(1/u)du.
5. Integrate: (1/2)ln|u| + C.
6. Substitute back u = x^2+4: (1/2)ln|x^2+4| + C.
Using trigonometric substitution:
1. Let's start by substituting x = 2tanθ. Therefore, dx = 2sec^2θdθ.
2. Substitute x and dx in the integral to get: ∫((2tanθ)/(4tan^2θ+4))(2sec^2θdθ).
3. Simplify the integral to: (4/4)∫(tanθ/(tan^2θ+1))sec^2θdθ.
4. Simplify further: ∫(tanθ/(sec^2θ))(sec^2θdθ).
5. Cancel out sec^2θ terms: ∫tanθ dθ.
6. Integrate: -ln|cosθ| + C.
7. Substitute back cosθ = 2/x: -ln|2/x| + C = -ln|2x^{-1}| + C = ln|x^2| + C.
After going through both solutions, we can see that there was a sign error in the trigonometric substitution solution. Instead of -ln|2/x|, it should have been ln|2/x|. Therefore, the correct integral using trigonometric substitution is ln|x^2| + C.
To summarize:
- Using u-substitution: the correct answer is (1/2)ln|x^2+4| + C.
- Using trigonometric substitution: the correct answer is ln|x^2| + C.
Please let me know if anything is unclear or if you have any further questions!