**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.