91Ó°ÊÓ

Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. $$\int_{0}^{1} 2 \sqrt{x^{2}+1} d x$$

Evaluate the integrals. Some integrals do not require integration by parts. $$\int\left(1+2 x^{2}\right) e^{x^{2}} d x$$

Solve the initial value problems in Exercises \(53-56\) for \(y\) as a function of \(x\). $$x \frac{d y}{d x}=\sqrt{x^{2}-4}, \quad x \geq 2, \quad y(2)=0$$

Evaluate the integrals in Exercises \(39-54\) $$\int \frac{\sqrt{1+\sqrt{x}}}{x} d x$$

Evaluate the integrals by making a substitution (possibly trigonometric) and then applying a reduction formula. $$\int_{0}^{\sqrt{3} / 2} \frac{d y}{\left(1-y^{2}\right)^{5 / 2}}$$

Evaluate the integrals. Some integrals do not require integration by parts. $$\int \frac{x e^{x}}{(x+1)^{2}} d x$$

Evaluate the integrals $$\int \sin 2 x \cos 3 x \, d x$$

Solve the initial value problems in Exercises \(53-56\) for \(y\) as a function of \(x\). $$\sqrt{x^{2}-9} \frac{d y}{d x}=1, \quad x>3, \quad y(5)=\ln 3$$

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{0}^{\infty} \frac{d \theta}{1+e^{\theta}}$$

Solve the initial value problems in Exercises \(53-56\) for \(y\) as a function of \(x\). $$\left(x^{2}+4\right) \frac{d y}{d x}=3, \quad y(2)=0$$