**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}.