Set up an integral to find the arclength of the curve y = ln(x) over the interval [1,5]. Use an appropriate trig substitution and then use the provided tables to complete the integration. Give the arclength to 2 decimal places.

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A: We will find out the required values.

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A:

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A: Consider the given integral. Find the u and v’.

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A: It is given that, ∫e2xcos3x dxWe have to evaluate it.

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A: Given function f(t) = eat sin(bt)

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A: First do some simplification of given relation than use the chain rule of derivative and simplify.

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A: Given:y=arcsinh(2x) and x0=6To find:The linear approximation to y=arcsinh(2x) at x0=6

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A: Since you have asked multiple questions in single request  so we  will be answering only first…

Q: Differentiate the following function.   r(t)=(t e^-t,5tln t,t sin t)

A:

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.