/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Find the integral. $$ \int \... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 20

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

Short Answer

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

Step by step solution

01

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 \( dt = -du / (2t) \).
02

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.
03

Evaluate the integral

Integrate the above expression to get \[ -\frac{1}{2} \cdot 2\sqrt{u} = -\sqrt{u} \].
04

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: \( -\sqrt{1 - t^2} + C \).

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