What is the integral of (cos(2sqrt(x)))/(sqrt(x)) ?

The integral of (cos(2sqrt(x)))/(sqrt(x)) is sin(2sqrt(x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (cos(2sqrt(x)))/(sqrt(x))

\int \frac{cos ( 2 x )}{x} d x

sin (2 x) + C

Solution steps

\int \frac{cos ( 2 x )}{x} d x

Rewrite using trig identities

= \int \frac{cos ^{2} ( x ) - sin ^{2} ( x )}{x} d x

Expand

\frac{cos ^{2} ( x ) - sin ^{2} ( x )}{x} : \frac{cos ^{2} ( x )}{x} - \frac{sin ^{2} ( x )}{x}

= \int \frac{cos ^{2} ( x )}{x} - \frac{sin ^{2} ( x )}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{cos ^{2} ( x )}{x} d x - \int \frac{sin ^{2} ( x )}{x} d x

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

\int \frac{sin ^{2} ( x )}{x} d x = x - \frac{1}{2} sin (2 x)

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

Simplify

x + \frac{1}{2} sin (2 x) - (x - \frac{1}{2} sin (2 x)) : sin (2 x)

= sin (2 x)

Add a constant to the solution

= sin (2 x) + C

\int 8 cos (θ) cos^{5} (sin (θ)) d θ

\frac{d}{d θ} (16 sin (θ))

\frac{d}{d x} (\frac{1}{5 x + 2})

(ln (3 x))^{2}

\int_{- 1}^{3} x^{3} + 2 x - 5 d x

What is the integral of (cos(2sqrt(x)))/(sqrt(x)) ?
The integral of (cos(2sqrt(x)))/(sqrt(x)) is sin(2sqrt(x))+C