a2f, a2f Q1: A- Show that + əy? ду? = 0, where f(x, y) = In(x² + y²) + tan-() %3D B- Evaluate the integral 나c dz dy dx 1-x² 3-x-y 1-x2.

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Q: integral by part x² dx

A: Given that  The integral is

Q: 2. dx Ix Also, Prove that 22 %3D dxdj = dy dX

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Q: Q2:A/ Evaluate the integrals 1 sec²(eVx + 1) dx J Vxe-V

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A:

Q: xP (2) X?-メ-3

A: 1. By Limit comparison test take ∑n=1∞Vn=∑1x32 .And limn→∞UnVn= finite.

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Q: Show that d/dz(z^2 z ̅) does not exist anywhere.

A: Here we show that ddzz2z does not exist anywhere.

Q: 1. √(1-x dx X

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

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A: We need to evaluate the integral ∫2x(x2+7)16dx with the help of substitution. This can be evaluated…

Q: Il-Find the following integrals by suitable substitutions 1 - x. (9 + x2)) 18.dx 2- 2XVX2 + 10.dx

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Q: оне dx 2 X√1+Lu² (x²) Secx sin 3x d x ۱/۱ ✓ で [H T =T TH ☑ M

A: Step 1:Step 2:

Q: sinh x 20. 1x1-3 A) 2. B)-2. D) 0.

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.