Recitation 1

Example 1: Evaluate $\int \sqrt{2x+1}dx$.

Solution: Let $u=2x+1$. Then $\int \sqrt{2x+1}dx=\int \sqrt{u}\frac{1}{2}du=\frac{1}{3}u^{3/2}+C=\frac{1}{3}(2x+1)^{3/2}+C$.

Example 2: Find $\int\frac{x}{\sqrt{1-4x^2}}dx$.

Solution: Let $u=1-4x^2$. Then $\int\frac{x}{\sqrt{1-4x^2}}dx=-\frac{1}{8}\int\frac{1}{\sqrt{u}}=-\frac{1}{8}(2\sqrt{u})+C=-\frac{1}{4}\sqrt{1-4x^2}+C$.

Example 3: Calculate $\int e^{5x}dx$.

Solution: Let $u=5x$. then $\int e^{5x}dx=\frac{1}{5}\int e^u du = \frac{1}{5}e^u+C=\frac{1}{5}e^{5x}+C$.

Problem 4: Evaluate $\int \sec^2 2\theta d\theta$.

Hint: Let $u=2\theta$.

Problem 5: Evaluate $\int \frac{dx}{5-3x}$.

Hint: Let $u=5-3x$.

Problem 6: Evaluate $\int \frac{(\ln x)^2}{x}dx$.

Hint: Let $u=\ln x$.

Problem 7: Evaluate $\int \sec^2\theta\tan^3\theta d\theta$.

Hint: Let $u=\tan\theta$.

Problem 8: Evaluate $\int e^x\sqrt{1+e^x} dx$.

Hint: Let $u=1+e^x$.

Problem 9: Evaluate $\int \sqrt{\cot x}\csc^2x dx$.

Hint: Let $u=\cot x$.

Problem 10: Evaluate $\int \frac{1+x}{1+x^2}dx$.

Hint: Split the integral into $\int \frac{1}{1+x^2}dx$ and $\int\frac{x}{1+x^2}dx$. For the second integral, use substitution $u=1+x^2$.

Problem 11: Evaluate $\int_1^2 \frac{e^{1/x}}{x^2}dx$.

Hint: Let $u=1/x$.

Problem 12: Evaluate $\int_0^a x\sqrt{a^2-x^2}dx$.

Hint: Let $u=a^2-x^2$.

Problem 13: Evaluate $\int_e^{e^4}\frac{dx}{x\sqrt{\ln x}}$.

Hint: Let $u=\ln x$.