- #1
Sczisnad
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Homework Statement
integral ln(2x+1)dx
Homework Equations
N/A
The Attempt at a Solution
I tried integration by parts,
Let u = ln(2x+1), dv = dx, du = 2/(2x+1)dx, v = x
ln(2x+1)dx = ln(2x+1)*x-(integral)x*(2/(2x+1))dx
So far, so good. The last integral can be turned into a simpler one by dividing 2x by 2x + 1, using polynomial long division. If you don't know that technique, it works out to 1 + -1/(2x + 1) in this problem.Sczisnad said:Homework Statement
integral ln(2x+1)dx
Homework Equations
N/A
The Attempt at a Solution
I tried integration by parts,
Let u = ln(2x+1), dv = dx, du = 2/(2x+1)dx, v = x
ln(2x+1)dx = ln(2x+1)*x-(integral)x*(2/(2x+1))dx
Evaluating the integral means finding the exact numerical value of a definite integral, which represents the area under a curve in a given interval. It is a way to measure the accumulation of a quantity over a certain range.
To evaluate an integral, you can use various techniques such as substitution, integration by parts, or the fundamental theorem of calculus. You can also use numerical methods like the trapezoidal rule or Simpson's rule.
A definite integral has specific limits of integration, which means it represents a specific area under a curve. An indefinite integral has no limits and represents a general antiderivative of a function.
No, not all integrals can be evaluated analytically. Some integrals have no closed form solution and can only be approximated using numerical methods.
Evaluating integrals is essential in many fields of science, such as physics, engineering, and economics. It allows us to calculate important quantities such as displacement, velocity, work, and profit, which are crucial for understanding real-world phenomena and making predictions.