Recitation 9

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 $u=\sqrt{4x^2+1}$.