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

The integral of 2cos(pi/3 x) is 6/pi sin((pix)/3)+C

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integral of 2cos(pi/3 x)

\int 2 cos (\frac{π}{3} x) d x

\frac{6}{π} sin (\frac{π x}{3}) + C

Solution steps

\int 2 cos (\frac{π}{3} x) d x

Take the constant out:

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

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

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

Apply u-substitution

: \int cos (u) \frac{3}{π} d u

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

Take the constant out:

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

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

Use the common integral:

\int cos (u) d u = sin (u)

= 2 \cdot \frac{3}{π} sin (u)

Substitute back

u = \frac{π x}{3}

= 2 \cdot \frac{3}{π} sin (\frac{π x}{3})

Simplify

2 \cdot \frac{3}{π} sin (\frac{π x}{3}) : \frac{6}{π} sin (\frac{π x}{3})

= \frac{6}{π} sin (\frac{π x}{3})

Add a constant to the solution

= \frac{6}{π} sin (\frac{π x}{3}) + C

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What is the integral of 2cos(pi/3 x) ?
The integral of 2cos(pi/3 x) is 6/pi sin((pix)/3)+C