int_(0)^(2)(dx)/(x^(2)-2x+2)

Identify integral: Identify the integral to be solved. We need to evaluate the integral of the function f(x) = 1/(x^2 - 2x + 2) from 0 to 2.Complete square: Complete the square for the denominator. The denominator x^2 - 2x + 2 can be written as (x - 1)^2 + 1 by completing the square.Substitute u: Substitute u for x - 1. Let u = x - 1, then du = dx. The limits of integration also change. When x = 0, u = -1, and when x = 2, u = 1.Rewrite in terms of u: Rewrite the integral in terms of u. The integral becomes ∫ from -1 to 1 of 1/(u^2 + 1) du.Recognize standard form: Recognize the standard integral form. The integral of 1/(u^2 + 1) du is a standard form whose antiderivative is arctan(u).Evaluate integral: Evaluate the integral. The integral from -1 to 1 of 1/(u^2 + 1) du is arctan(u) evaluated from -1 to 1.Calculate definite integral: Calculate the definite integral. The value of the integral is arctan(1) - arctan(-1) which is π/4 - (-π/4) = π/2.

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$\int_{0}^{2}\frac{dx}{x^{2}-2x+2}$

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Calculus

Evaluate definite integrals using the chain rule

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\int_{0}^{2}\frac{dx}{x^{2}-2x+2}

Identify integral: Identify the integral to be solved. $\newline$ We need to evaluate the integral of the function $f(x) = \frac{1}{x^2 - 2x + 2}$ from $0$ to $2$ .
Complete square: Complete the square for the denominator. $\newline$ The denominator $x^2 - 2x + 2$ can be written as $(x - 1)^2 + 1$ by completing the square.
Substitute $u$ : Substitute $u$ for $x - 1$ . $\newline$ Let $u = x - 1$ , then $du = dx$ . The limits of integration also change. When $x = 0$ , $u = -1$ , and when $x = 2$ , $u = 1$ .
Rewrite in terms of $u$ : Rewrite the integral in terms of $u$ . The integral becomes $\int_{-1}^{1} \frac{1}{u^2 + 1} \, du$ .
Recognize standard form: Recognize the standard integral form. $\newline$ The integral of $\frac{1}{u^2 + 1}$ $du$ is a standard form whose antiderivative is $\text{arctan}(u)$ .
Evaluate integral: Evaluate the integral. $\newline$ The integral from $-1$ to $1$ of $\frac{1}{u^2 + 1}$ du is $\text{arctan}(u)$ evaluated from $-1$ to $1$ .
Calculate definite integral: Calculate the definite integral. $\newline$ The value of the integral is $\arctan(1) - \arctan(-1)$ which is $\frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{2}$ .