Please use excel. Determine the approximate integration using simple and modified trapezoidal, 1/3 Simpson's and 3/8 Simpson's Rule. ⌠41cosx (0.25x2+1.90)dx, use n=3

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A: I = ∫ x ln (5+x) dxLet  5 + x = u       dx  = duand,          x = 4 - 5

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Q: Use integration by parts to prove the reductionformula.

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Q: Evaluate the integral. (Use C for the constant of integration.) 45x-5 dx

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Q: First make a substitution and then use integration by parts to evaluate the integral. dx

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Q: Evaluate the integral. (Use C for the constant of integration.) (3* – 5*) dx

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Q: Evaluate the integral. (Use C for the constant of integration.) dx 5x(1 +5x)

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Q: Evaluate the integral. (Use C for the constant of integration.) (7 – 17x) dx

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Q: Evaluate the integral. (Use C for the constant of integration.) (3× – 5*) dx

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Q: J 2 x sin -1 xdx

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Q: Evaluate the integral. (Use C for the constant of integration.) 25x-5 dx

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Q: Evaluate the integral. (Use C for the constant of integration.) (3x5 + 1) dx

A: see 2nd step.

Q: Evaluate the integral. (Use C for the constant of integration.) 9e 9x dx

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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.