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utkarshakash
Gold Member
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Homework Statement
[itex]\int \left( \dfrac{ln(1+\sqrt[6]{x})}{\sqrt[3]{x} + \sqrt{x}} \right) dx [/itex]
The Attempt at a Solution
Let x^(1/6) = t
[itex]\int \dfrac{ln(1+t)t^3}{1+t} dt [/itex]
Simon Bridge said:You want to evaluate:
$$\int \left( \dfrac{\ln(1+\sqrt[6]{x})}{\sqrt[3]{x} + \sqrt{x}} \right) dx$$
You attempted it by doing the substitution: ##x^{(1/6)} = t## - which gets you:
$$\int \dfrac{\ln(1+t)t^3}{1+t} dt$$
... and, from there, you get stuck?
Are you missing a factor of 6 in there?
Have you tried putting u=1+t ?
Evaluating an integral means to find the numerical value of the integral, which represents the area under a curve in a specific interval.
To evaluate an integral, you can use various techniques such as substitution, integration by parts, or trigonometric identities. First, identify the appropriate method to use, then apply the necessary steps to solve the integral.
Limits of integration are the upper and lower bounds that define the interval in which the integral is being evaluated. These limits can be numbers, variables, or infinity.
No, not every integral can be evaluated analytically. Some integrals are considered to be unsolvable, and in those cases, numerical methods such as the trapezoidal rule or Simpson's rule can be used to approximate the value of the integral.
Evaluating integrals is crucial in many areas of science, such as physics, engineering, and economics. It allows us to calculate important quantities such as displacement, velocity, and area, which are essential in understanding real-world phenomena and making accurate predictions.