Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant...

Question:

Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant of integration.)

{eq}\int \frac{x^3}{\sqrt{x^2 + 64}}\;dx,\;\;x=8\,tan\,\theta {/eq}

Trigonometric Substitution:

Trigonometric substitution is mainly used to remove radicals from integrands so the integral can be evaluated.

For arguments of the form,

$$a^2 - u^2, \hspace{1 cm} \text{let } u = a \sin \theta$$ $$u^2 - a^2, \hspace{1 cm} \text{let } u = a \sec \theta$$ $$u^2 + a^2, \hspace{1 cm} \text{let } u = a \tan \theta.$$

Since we will be utilizing {eq}\tan \theta{/eq}, let's review a couple of items.

a) The Pythagorean identity {eq} \tan^2 \theta + 1 = \sec^2 \theta{/eq}.

b) The derivatives of secant and tangent.

$$\frac{d}{d\theta} \sec \theta = \sec \theta \tan \theta$$ $$ \frac{d}{d\theta} \tan \theta = \sec^2 \theta$$

Answer and Explanation: 1

Become a Study.com member to unlock this answer!

View this answer

To evaluate the integral {eq}\int \frac{x^{3}}{\sqrt{x^2 + 64}}\, dx {/eq}, we first let {eq} x= 8 \tan \theta{/eq}.

Then {eq} dx = 8 \sec^2 \theta...

See full answer below.


Learn more about this topic:

Loading...
How to Use Trigonometric Substitution to Solve Integrals

from

Chapter 13 / Lesson 12
15K

Trigonometric substitutions can be useful by plugging in a function of a variable, thus simplifying the calculation of an integral. Learn how to solve integrals using substitution, tables, by parts, and Riemann Sums through a variety of examples.


Related to this Question

Explore our homework questions and answers library