How to determine whether a trig integral is True or False?

Question

For the following 3 questions, I know that the answer is false, but I'm not sure about how to provide a counter example. Can anyone please help me out?

Determine whether the following integrals are true or false. Provide a counter example if false and give an explanation if true:

a) If $x = 4\tan\theta$, then $\csc\theta = \frac{4}{x}$

b) The integral $\int_{1}^{2}\sqrt{x^2-1} dx$ does not have a real finite value

c) The integral $\int\frac{1}{x^2+4x+9} dx$ cannot be evaluated using trigonometric substitution

Answer 1

a) This isn't really an integral here, but a (purported) identity. It claims that, given any $x$ and $\theta$ such that $x = 4\tan\theta$, then $\csc \theta = 4/x$ is also true. Make a counter-example by finding some specific value of $x$ and $\theta$ such that $x = 4\tan\theta$ is true, but $\csc \theta \neq 4/x$.

It shouldn't be too hard. Just choose a value of $\theta$, and let $x = 4 \tan \theta$. Most values of $\theta$ will produce a counter-example.

b) For this one, you can either compute the integral or just appeal to integration theorems. The fact that $\sqrt{x^2 - 1}$ is real-valued and continuous on $[1, 2]$ implies that it has some finite value. This is not strictly a counter-example though!

c) Provide a trigonometric substitution. A good start is to complete the square: $$\frac{1}{x^2 + 4x + 9} = \frac{1}{x^2 + 4x + 4 + 5} = \frac{1}{(x + 2)^2 + 5}.$$

Answer 2

For b, notice that since $\sqrt{x^2-1}$ is monotonically increasing (from $1$ to $\infty$), and $\sqrt{2^2-1} < 2$, then $\int_1^2 \sqrt{x^2-1} dx < \int_1^2 2 dx$.

Therefore the area is less than $2$, and is definitely finite.

Answer 3

a) If $ x = 4\tan\theta,$ then $\csc\theta = \frac 4x$ is not always true, because it implies $$ \tan\theta = \frac x4 = \sin\theta $$ which is not always true.

b) $$\int_{1}^{2}\sqrt{x^2-1} dx$$ is a real finite number because the integrand is continuous and the interval is bounded.

c) $$\int\frac{1}{x^2+4x+9} dx$$ can be evaluated using trigonometric substitution. complete the square and let $(x+2) = \sqrt 5 \tan (t)$