The file contains several mathematical problems involving summation, integration, and approximation techniques. Here are the specific tasks listed for each problem:

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Question 1

1. (a) Express the given sum as a function of $n$ and evaluate it for $n = 10$.
2. (b) Determine the value of the sum as $n \to \infty$.
3. (c) Derive an expression for the area under $f(x) = \cos x$ on $[0, b]$, using the right endpoint, as a limit.
4. (d) Estimate the area for $b = \pi/2$ using four subintervals and the midpoint rule.

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Question 2

1. (a) Evaluate $\int_{1}^{3}(3x^2 + 2x + 1)dx$ as a limit of an upper Riemann sum.
2. (b) Evaluate $\int_{0}^{1}\sqrt{x}dx$ using a Riemann sum limit.
3. (c) Compute $\int_{1}^{2}x dx$ via a Riemann sum limit.
4. (d) Calculate $\sum_{i=1}^{100} (i + 1)$.

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Question 3

1. (a) Using the Fundamental Theorem of Calculus, evaluate the derivatives of:
   • $\frac{d}{dx} \int_{2x}^{x^2} \cos(5t) dt$,
   • $\frac{d}{du} \int_{-3}^{u} \frac{1}{t^2 + 1} dt$,
   • $\frac{d}{dx} \int_{x}^{5} \sin(t^2) dt$,
   • $\frac{d}{dx} \int_{x}^{2x^3} \cos t dt$.
2. (b) Approximate $\int_{0}^{4}(x^2 - x) dx$ using a Riemann sum with 4 subintervals (right endpoints).
3. (c) Find a function $f(t)$ and a number $a$ such that $6 + \int_{a}^{t^2} f(t) dt = 2\sqrt{x}$, $\forall x > 0$.

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Question 4

1. (a) Estimate $\int_{2}^{3} \frac{1}{x+1} dx$ using Simpson’s Rule ($n=4$).
2. (b) Compute its exact value and the error.
3. (c) Approximate $\int_{0}^{1} \sqrt{2x + 1} dx$ using the trapezoidal scheme ($h=0.25$).
4. (d) Apply the midpoint rule to estimate $\int_{0}^{1} x^2 dx$ with $h=0.25$.

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Question 5

Write down the following integrals:

• $\int x^n dx$,
• $\int \frac{1}{x} dx$,
• $\int e^x dx$,
• $\int a^x dx$,
• $\int \sin x dx$,
• $\int \cos x dx$,
• $\int \sec^2 x dx$,
• $\int \csc^2 x dx$,
• $\int \sec x \tan x dx$,
• $\int \csc x \cot x dx$.

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Question 6

1. (a) Using integration by parts, evaluate:
   • $\int xe^x dx$,
   • $\int xe^x \sin x dx$.
2. (b) Evaluate definite integrals using integration by parts:
   • $\int_{0}^{1} \tan^{-1} x dx$,
   • $\int_{2}^{4} x^2 e^x dx$.

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5 Related Questions

1. How is the midpoint rule applied to approximate definite integrals?
2. What is the difference between Riemann sums and Simpson’s rule?
3. How can the Fundamental Theorem of Calculus simplify integration tasks?
4. What is the role of limits in defining integrals and summations?
5. Why is integration by parts effective for products of functions?

Tip

Always organize problems with clear annotations and diagrams (if necessary) to enhance understanding and accuracy during problem-solving.