What is the integral of 1/(x^2)cos(1/(sqrt(x))) ?

The integral of 1/(x^2)cos(1/(sqrt(x))) is -2((sin(1/(sqrt(x))))/(sqrt(x))+cos(1/(sqrt(x))))+C

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integral of 1/(x^2)cos(1/(sqrt(x)))

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

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

Solution steps

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

Apply u-substitution

: \int \frac{2 cos ( \frac{1}{u} )}{u ^{3}} d u

= \int \frac{2 cos ( \frac{1}{u} )}{u ^{3}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{cos ( \frac{1}{u} )}{u ^{3}} d u

Apply u-substitution

: \int - v cos (v) d v

= 2 \cdot \int - v cos (v) d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 (- \int v cos (v) d v)

Apply Integration By Parts

: v sin (v) - \int sin (v) d v

= 2 (- (v sin (v) - \int sin (v) d v))

Use the common integral:

\int sin (v) d v = - cos (v)

= 2 (- (v sin (v) - (- cos (v))))

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 1/(x^2)cos(1/(sqrt(x))) ?
The integral of 1/(x^2)cos(1/(sqrt(x))) is -2((sin(1/(sqrt(x))))/(sqrt(x))+cos(1/(sqrt(x))))+C