int_(-1)^(1)(1)/(2-x)dx Steps Graph

Identify integral: Identify the integral to be solved. We need to evaluate the integral of the function (1)/(2-x) from -1 to 1.Recognize form: Recognize the form of the integral. The integral is of the form ∫(1)/(a-x)dx, which is a standard form and its antiderivative is -ln|a-x| + C, where C is the constant of integration.Calculate indefinite integral: Calculate the indefinite integral. The indefinite integral of (1)/(2-x) is -ln|2-x| + C.Evaluate definite integral: Evaluate the definite integral. We need to evaluate the integral from -1 to 1, which means we will substitute these values into our antiderivative and calculate the difference. ∫(-1 to 1)(1)/(2-x)dx = [-ln|2-x|] from -1 to 1 = -ln|2-1| + ln|2+1| = -ln|1| + ln|3| Since ln|1| is 0, the expression simplifies to ln|3|.Check for discontinuities: Check for any discontinuities within the interval. The function (1)/(2-x) has a discontinuity at x=2, which is not within the interval from -1 to 1. Therefore, there are no issues with discontinuities affecting the integral.

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$\int_{-1}^{1}\frac{1}{2-x}dx$

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Calculus

Evaluate definite integrals using the chain rule

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\int_{-1}^{1}\frac{1}{2-x}dx

Identify integral: Identify the integral to be solved. $\newline$ We need to evaluate the integral of the function $\frac{1}{2-x}$ from $-1$ to $1$ .
Recognize form: Recognize the form of the integral. $\newline$ The integral is of the form $\int\frac{1}{a-x}\,dx$ , which is a standard form and its antiderivative is $-\ln|a-x| + C$ , where $C$ is the constant of integration.
Calculate indefinite integral: Calculate the indefinite integral. $\newline$ The indefinite integral of $\frac{1}{2-x}$ is $-\ln|2-x| + C$ .
Evaluate definite integral: Evaluate the definite integral. $\newline$ We need to evaluate the integral from $-1$ to $1$ , which means we will substitute these values into our antiderivative and calculate the difference. $\newline$ $\int_{-1}^{1}\frac{1}{2-x}\,dx = [-\ln|2-x|] \text{ from } -1 \text{ to } 1$ $\newline$ $= -\ln|2-1| + \ln|2+1|$ $\newline$ $= -\ln|1| + \ln|3|$ $\newline$ Since $\ln|1|$ is $0$ , the expression simplifies to $\ln|3|$ .
Check for discontinuities: Check for any discontinuities within the interval. The function $(1)/(2-x)$ has a discontinuity at $x=2$ , which is not within the interval from $-1$ to $1$ . Therefore, there are no issues with discontinuities affecting the integral.