Problem 20 - Essential Calculus

Question

Find the integral. $$ \int \frac{t}{\left(1-t^{2}\right)^{3 / 2}} d t $$

Step-by-Step Solution

Answer

The integral of \( \frac{t}{\left(1-t^{2}\right)^{3 / 2}} dt \) is \( -\sqrt{1 - t^2} + C \).

Step 1: Substitution

Let's start by letting \( u = 1 - t^2 \). Then, differentiate \( u \) with respect to \( t \) to find \( du \). We have \( du = -2t dt \). This gives us \( dt = -du / (2t) \).

Step 2: Rewrite the integral

Replace \( t^2 \) and \( dt \) in the integral with \( u \) and \( du \) respectively and simplify. This gives us: \[ \int \frac{t}{(u)^{3/2}} \cdot -du/(2t) = -\frac{1}{2} \int \frac{1}{\sqrt{u}} du \]. Now our integral is in a form that we can readily integrate.

Step 3: Evaluate the integral

Integrate the above expression to get \[ -\frac{1}{2} \cdot 2\sqrt{u} = -\sqrt{u} \].

Step 4: Back substitution

Substitute back \( u \) into the expression from step 1 to find the result of the original integral: \( -\sqrt{1 - t^2} \).Integral of an integral is an antiderivative, so to indicate this we usually add a constant (\( C \)) at the end. Therefore, the answer will be: \( -\sqrt{1 - t^2} + C \).