Evaluate [math]\int \sqrt{a^2-x^2} \; dx[/math] using equations, and compare your answer with that found using equations.

• lab{7.3.2a} Write a set of equations for integrating functions of [math]\sqrt{a^2+x^2}[/math] which are analogous to equations, but are based on the identity [math]1+\cot^2\theta = \csc^2\theta[/math].
• Select an interval to which [math]\theta[/math] can be restricted so that it is uniquely determined by the equations in part \ref{ex7.3.2a} and so that [math]x[/math] can take on all real number values.

Evaluate [math]\int \frac{dx}{\sqrt{a^2+x^2}}[/math] using the subtraction described in Problem Exercise.

What is the set to which [math]\theta[/math] should be restricted if the substitution of [math]|a| \csc \theta[/math] for [math]x[/math] makes [math]\sqrt{x^2-a^2}[/math] equal to [math]|a| \cot \theta[/math], defines [math]\theta[/math] unambiguously, and also lets [math]x[/math] take on all real values such that [math]|x| \geq |a|[/math]?

Evaluate the following integrals.

• [math]\int \frac{\sqrt{x^2-9}}x \; dx[/math]
• [math]\int \sqrt{(x^2-1)^3} \; dx[/math]
• [math]\int x\sqrt{16-x^2} \; dx[/math]
• [math]\int x^3\sqrt{x^2-4} \; dx[/math]
• [math]\int \frac{dx}{x^2-9}[/math]
• [math]\int \sqrt{(x^2+4)^3} \; dx[/math]
• [math]\int \frac{x \; dx}{\sqrt{(a^2+x^2)^3}}[/math]
• [math]\int \frac{x^3\;dx}{\sqrt{4-x^2}}[/math]
• [math]\int \sqrt{x^2-a^2} \; dx[/math]
• [math]\int \frac{dx}{\sqrt{(x^2-25)^3}}[/math].

Evaluate [math]2\int_{a-h}^a \sqrt{a^2-x^2} \; dx[/math], and hence find the area of a segment of height [math]h[/math] in a circle of radius [math]a[/math].

Evaluate the following definite integrals.

• [math]\int_0^4 \frac{dx}{\sqrt{9+x^2}}[/math]
• [math]\int_0^{\sqrt5} \frac{x\;dx}{\sqrt{4+x^2}}[/math]
• [math]\int_0^{\frac14} \frac{dx}{\sqrt{1-4x^2}}[/math]
• [math]\int_{\frac4{\sqrt3}}^4 \frac{dx}{\sqrt(x^2-4)^3}[/math]
• [math]\int_3^4 x\sqrt{25-x^2} \; dx[/math]
• [math]\int_{\frac52}^{\frac72} \frac{dx}{\sqrt{4x^2-9}}[/math].

Integrate

• [math]\int \frac{dx}{5+x^2}[/math]
• [math]\int \frac{dx}{2x^2+8}[/math]
• [math]\int \frac{dx}{x^2+2x+5}[/math]
• [math]\int \frac{dx}{2x^2+12x+20}[/math]
• [math]\int \frac{dx}{(2x^2+6)^2}[/math]
• [math]\int \frac{dx}{(x^2-4x+8)^2}[/math]
• [math]\int \frac{dx}{(x^2+9)^3}[/math]
• [math]\int \frac{dx}{(x^2+2x+2)^3}[/math].

By the substitutions used to change equation to and by the reduction formula,, verify the following reduction formula (where [math]b^2-4ac \lt 0[/math]):