Question

Why don’t we need to have a square root involved in order to apply trigonometric substitution with the tangent? In other words, why can we use the substitution $x=a\tan u$when we see $x^2+a^2$, even though we can’t use the substitution $x=a\sin u$unless the integrand involves the square root of$a^2-x^2$? (Hint: Think about domains.)

Short answer

Ans: The domain or the substitution of $x=\sin u$ makes sense only for $x\in [-1,1]$.

But the domain of the $a^2-x^2\text{is}(-\mathrm{\infty },\mathrm{\infty })$.

On the other hand, the substitution of $x=\tan u$ makes sense only for $x\in (-\mathrm{\infty },\mathrm{\infty })$. That is everywhere that $a^2+x^2$ is defined.

Step-by-step solution

Step 1. Given information.

given,

why can we use the substitution$x=a\tan u$when we see$x^2+a^2$

Step 2.  Solution:-

The domain or the substitution of $x=\sin u$ makes sense only for $x\in [-1,1]$.

But the domain of the $a^2-x^2\text{is}(-\mathrm{\infty },\mathrm{\infty })$.

On the other hand, the substitution of x=tan⁡u makes sense only for $x\in (-\infty ,\infty )$. That is everywhere that $a^2+x^2$ is defined