cos x + sin xdx.Split up fractions Evaluatesec x

To evaluate: ∫cosx+sin3xsecxdx. Given integrand can be written as:…

Answered: cos x + sin x dx. Split up fractions…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

cos x + sin x
dx.
Split up fractions Evaluate
sec x

Expert Solution

Step 1

To evaluate: $\int \frac{\cos x + \sin^{3} x}{s e c x} d x$ .

Given integrand can be written as:

$\begin{array}{rcl} \int \frac{\cos x + \sin^{3} x}{s e c x} d x & = & \int [\frac{\cos x}{s e c x} + \frac{\sin^{3} x}{s e c x}] d x \\ = & \int \cos^{2} x d x + \int \sin^{3} x \cos x d x \end{array}$

Let $I_{1} = \int \cos^{2} x d x$ and $I_{2} = \int \sin^{3} x \cos x d x$

Now, solving $I_{1}$ .

$\begin{array}{rcl} I_{1} & = & \int \cos^{2} x d x \\ = & \int (\frac{1 + \cos 2 x}{2}) d x \\ = & \frac{1}{2} \int (1 + \cos 2 x) d x \\ = & \frac{1}{2} (x + \frac{\sin 2 x}{2}) + C_{1} \end{array}$

Step 2

Now, solving $I_{2}$ .

$I_{2} = \int \sin^{3} x \cos x d x$

Let $t = \sin x$

$\begin{array}{rcl} \frac{d t}{d x} & = & \cos x \\ d t & = & \cos x d x \end{array}$

$\begin{array}{rcl} I_{2} & = & \int \sin^{3} x \cos x d x \\ = & \int t^{3} d t \\ = & \frac{t^{4}}{4} + C_{2} \\ = & \frac{\sin^{4} x}{4} + C_{2} \end{array}$