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.

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