Example 1: Use the Simpson’s Rule with n=10 to approximate \int_1^2\frac{1}{x}dx.
Hint: S_n = \frac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+\ldots+4f(x_{n-1})+f(x_n)\right].
Example 2: How large should we take n in order to guarantee that the Simpson’s Rule approximation for \int_1^2\frac{1}{x}dx is accurate to within 0.0001?
Solution: Because f^{(4)}(x)=24/x^5 and it is at most 24 for 1\leq x\leq 2, we can bound |E_S| by 24(2-1)^5/180n^4. It suffices to find n such that 24(2-1)^5/180n^4 \leq 0.0001. Solve the inequality and get n > 6. Notice that n must be even. We should take n=8.
Problem 3: Use the Trapezoidal Rule, the Midpoint Rule, the Simpson’s Rule to approximate \int_1^2\sqrt{x^3-1}dx with n=10.
Hint: T_{10} = \frac{\Delta x}{2}(\sqrt{1^3-1}+2\sqrt{1.1^3-1}+\ldots+2\sqrt{1.9^3-1}+\sqrt{2^3-1}). M_{10} = \Delta x(\sqrt{1.05^3-1}+\sqrt{1.15^3-1}+\ldots+\sqrt{1.95^3-1}). S_{10}=\frac{\Delta x}{3}(\sqrt{1^3-1}+4\sqrt{1.1^3-1}+2\sqrt{1.2^3-1}+\ldots+4\sqrt{1.9^3-1}+\sqrt{2^3-1}).
Problem 4: Evaluate \int e^{2x}\sin{3x}dx.
Hint: Use integral by parts twice.
Problem 5: Evaluate \int\frac{x^3+4}{x^2+4}dx.
Hint: Long division and split the integral.
Problem 6: Evaluate \int\frac{e^{2x}}{e^{2x}+3e^x+2}dx.
Hint: Use u = e^x and partial fraction.
Problem 7: Evaluate \int\frac{1}{x\sqrt{4x+1}}dx.
Hint: Use u=\sqrt{4x+1}.
Problem 8: Evaluate \int\frac{1}{x\sqrt{4x^2+1}}dx.
Hint: Use u=\sqrt{4x^2+1}.