"Evaluate the following integrals. ∫cos210xdx" was answered by our experts

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Shirley Thompson

Answered question

2021-12-14

Evaluate the following integrals.

\int \cos^{2} 10 x d x

Answer & Explanation

einfachmoipf

Beginner2021-12-15Added 32 answers

Step 1
Given:

\int \cos^{2} 10 x d x

we know that

1 + \cos (20) = 2 \cos^{2} 0

similarly,

2 \cos^{2} (10 x) = 1 + \cos (20 x)

\cos^{2} (10 x) = \frac{1 + \cos (20 x)}{2}

now replacing

\cos^{2} (10 x)

with

\frac{1 + \cos (20 x)}{2}

in given integral
Step 2
so,

\int \cos^{2} (10 x) d x = \int \frac{1 + \cos (20 x)}{2} d x

= \int (\frac{1}{2} + \frac{\cos (20 x)}{2}) d x

= \frac{1}{2} \int d x + \frac{1}{2} \int \cos (20 x) d x

(∵ \int d x = x + x, \int \cos (k x) d x = \frac{\sin (k x)}{k} + c)

= \frac{1}{2} (x) + \frac{1}{2} (\frac{\sin (20 x)}{20}) + c

= \frac{x}{2} + \frac{\sin (20 x)}{40} + c

hence, given integral is equal to

\frac{x}{2} + \frac{\sin (20 x)}{40} + c

yotaniwc

Beginner2021-12-16Added 34 answers

$\int {\cos (10 x)}^{2} d x$
$\int \frac{\cos (t)^{2}}{10} d t$
$\frac{1}{10} \times \int \cos (t)^{2} d t$
$\frac{1}{10} \times \int \frac{1 + \cos (2 t)}{2} d t$
$\frac{1}{10} \times \frac{1}{2} \times \int 1 + \cos (2 t) d t$
$\frac{1}{20} \times (\int 1 d t + \int \cos (2 t) d t)$
$\frac{1}{20} \times (10 x + \frac{\sin (2 \times 10 x)}{2})$
Simplify
$\frac{1}{2} x + \frac{\sin (20 x)}{40}$
Solution:
$\frac{1}{2} x + \frac{\sin (20 x)}{40} + C$

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