Question

Evaluate the integral. $ \displaystyle \int x \sqrt{1 - x^4}\ dx $

   Evaluate the integral.

$ \displaystyle \int x \sqrt{1 - x^4}\ dx $

Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart 8th Edition

Chapter 7, Problem 29

Instant Answer

Solved by Expert J Hardin

01/07/2018

Step 1

We can rewrite this as $du/2 = x dx$. This allows us to rewrite the integral in terms of $u$: \[ \int x \sqrt{1 - x^4}\ dx = \frac{1}{2} \int \sqrt{1 - u^2}\ du \] Show more…

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Evaluate the integral. $ \displaystyle \int x \sqrt{1 - x^4}\ dx $

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Key Concepts

U-Substitution

A technique used in integration where a substitution is made to simplify the integrand, often by letting a complicated expression equal a new variable, which transforms the integral into a more manageable form.

Integration of Radical Functions

This concept involves methods for integrating expressions with square roots, which may require substituting to obtain a standard form that can be solved using familiar antiderivatives.

Inverse Trigonometric Integration

Certain integrals, especially those involving a square root of a quadratic expression like sqrt(1 - u^2), naturally lead to antiderivatives expressed in terms of inverse trigonometric functions, making recognition of these forms key in solving the integral.

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Transcript

00:01 Let's evaluate the integral of x times the square root of 1 minus x to the 4th.

00:06 Before we go to a trig sub, we should try a u sub first.

00:09 Let's take u to be x squared so that du is 2x dx.

00:14 Equivalently, du over 2 is x dx, and we see those terms in the original integrand.

00:20 Here's x and dx.

00:23 So we can write this as 1 1⁄2, integral, radical 1 minus u squared, du.

00:34 And then now for this new integral, we can go ahead and use the tricks up.

00:43 So here we should take u to be 1 sine theta.

00:50 Then du is cosine theta.

00:59 So plugging this in, we have one half integral, 1 minus sine squared times cosine theta.

01:18 So for our next step, using pythagorean theorem, 1 minus sine squared is cosine squared.

01:33 And we know that the square root of cosine squared is just cosine.

01:43 So we have one half, cosine squared theta.

01:52 So now we have a trig integral cosine squared.

01:56 So for this, it'll be best to use the identity.

02:04 One plus cosine of 2 theta, all divided by 2.

02:08 Half angle identity.

02:11 Running out of room here, let me go to the next page.

02:15 Simplifying, we have 1 fourth, 1 plus cosine 2 ,000.

02:21 Theta.

02:29 Now we're ready to integrate...

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