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

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

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integral of 2cos(x/3)

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

6 sin (\frac{x}{3}) + C

Solution steps

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

Apply u-substitution

: \int cos (u) \cdot 3 d u

= 2 \cdot \int cos (u) \cdot 3 d u

Take the constant out:

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

= 2 \cdot 3 \cdot \int cos (u) d u

Use the common integral:

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

= 2 \cdot 3 sin (u)

Substitute back

u = \frac{x}{3}

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

Simplify

= 6 sin (\frac{x}{3})

Add a constant to the solution

= 6 sin (\frac{x}{3}) + C

f (x) = x^{2} - 6 x - 7

x y^{2} y^{^{'}} = y^{3} - 5 x^{3}

\frac{d}{d θ} (tan^{2} (θ))

y^{^{''}} + 11 y^{^{'}} + 24 y = 0, y (0) = 0, y^{^{'}} (0) = - 7

x \to 0 lim (\frac{( 9 ^{x} - 3 ^{x} )}{x})