Problem 3
Question
Find the indefinite integral. $$ \int \frac{1}{3-2 x} d x $$
Step-by-Step Solution
Verified Answer
The indefinite integral of \( \frac{1}{3-2x} dx \) is \(- \frac{1}{2}\ln|3-2x| + C\).
1Step 1: Identify the form of the function
The given function is in the form of \(1/(p*x)\), where \(p = -2\) and \(x = 3 - 2x\). The integral for this form is given by \(\int \frac{1}{p*x} dx = (1/p) ln |p*x|\). We should apply this formula to solve the integral.
2Step 2: Apply the integral formula
Applying the formula, we get \((1/-2) ln |3 - 2x|\) for the integral. The resulting integral is \(-1/2 * ln |3 - 2x|\).
3Step 3: Add the constant of integration
The final step in finding the indefinite integral is to add the constant of integration, usually denoted by 'C'. So, we write the final result as \(-1/2 * ln |3 - 2x| + C\).
Other exercises in this chapter
Problem 3
Graphical Reasoning use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \
View solution Problem 3
Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}
View solution Problem 3
Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal
View solution Problem 3
In Exercises \(1-6,\) find the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{k=0}^{4} \frac{1}{k^{2}+1} $$
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