∫√(x²+1)/(x+1)dx
plz show step i know it messy plz i really need it
This one looks easy, using trig substitutions, but it is not! Enter your function at the url below and it will show all the messy steps in the solution.
http://www.integral-calculator.com/
To integrate √(x²+1)/(x+1) with respect to x, we can use a substitution technique. Let's start with the steps to solve this integral step by step:
Step 1: Substitute a variable to simplify the integral: Let u = x + 1. This implies x = u - 1. Also, we need to find the derivative of u with respect to x, which is du/dx = 1.
Step 2: Express the integral in terms of the new variable u:
∫ (√(x²+1))/(x+1) dx
Substituting, we get:
∫ (√((u-1)² + 1))/u du
Simplifying:
∫ (√(u² - 2u + 1 + 1))/u du
∫ (√(u² - 2u + 2))/u du
Step 3: Separate the expression:
∫ (√(u² - 2u + 2))/u du = ∫ (√(u² - 2u + 2))/u du
Step 4: Decompose the fraction:
∫ (√(u² - 2u + 2))/u du = ∫ √(u² - 2u + 2)/√u du
Step 5: Now, we have a fraction where the numerator and denominator have different variables. Let's simplify it using another substitution:
Let v = u² - 2u + 2. This implies u² - 2u + 1 = v - 1. Also, dv/du = 2u - 2 ⇒ du = dv/(2u - 2).
Step 6: Express the integral in terms of the new variable v:
∫ √(u² - 2u + 2)/√u du = ∫ √(v - 1)/√u dv/(2u - 2)
Substituting the values we found:
∫ √(v - 1)/√u dv/(2u - 2)
= 1/2 ∫ √(v - 1)/√u dv
Step 7: Rewrite the expression in terms of a single variable:
Using the fact that v = u² - 2u + 2 and u = x + 1, we can write it as:
1/2 ∫ √(u² - 2u + 2)/√u du = 1/2 ∫ √(x² - x + 2)/√(x + 1) dx
Step 8: Simplify the expression further if possible, or proceed with numerical techniques to approximate the integral.
Apologies, but it seems that the integral ∫√(x²+1)/(x+1)dx cannot be further simplified analytically. To calculate an approximate value, you can use numerical methods like Simpson's rule, the trapezoidal rule, or computer software programs like Mathematica or Wolfram Alpha.