What is the integral of cos(2/(3x)) ?

The integral of cos(2/(3x)) is xcos(2/(3x))+2/3 \Si(2/(3x))+C

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integral of cos(2/(3x))

\int cos (\frac{2}{3 x}) d x

x cos (\frac{2}{3 x}) + \frac{2}{3} Si (\frac{2}{3 x}) + C

Solution steps

\int cos (\frac{2}{3 x}) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{2 cos ( v )}{v ^{2}} d v

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

Take the constant out:

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

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

Apply Integration By Parts

: - \frac{cos ( v )}{v} - \int \frac{sin ( v )}{v} d v

= \frac{1}{3} (- 2 (- \frac{cos ( v )}{v} - \int \frac{sin ( v )}{v} d v))

Use the nonelementary integral:

\int \frac{sin ( x )}{x} d x = Si (x)

= \frac{1}{3} (- 2 (- \frac{cos ( v )}{v} - Si (v)))

Substitute back

= \frac{1}{3} (- 2 (- \frac{cos ( \frac{2}{3 x} )}{\frac{2}{3 x}} - Si (\frac{2}{3 x})))

Simplify

\frac{1}{3} (- 2 (- \frac{cos ( \frac{2}{3 x} )}{\frac{2}{3 x}} - Si (\frac{2}{3 x}))) : x cos (\frac{2}{3 x}) + \frac{2}{3} Si (\frac{2}{3 x})

= x cos (\frac{2}{3 x}) + \frac{2}{3} Si (\frac{2}{3 x})

Add a constant to the solution

= x cos (\frac{2}{3 x}) + \frac{2}{3} Si (\frac{2}{3 x}) + C

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What is the integral of cos(2/(3x)) ?
The integral of cos(2/(3x)) is xcos(2/(3x))+2/3 \Si(2/(3x))+C