integral of cos^2(x/3) — step-by-step solution

1) Let's integrate:

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

We reduce the square using the identity

{\cos (\frac{x}{3})}^{2} = \frac{1 + \cos (2 \cdot \frac{x}{3})}{2}

ⅰ

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

\int \frac{1 + \cos (\frac{2 x}{3})}{2} d x

\int \frac{1 + \cos (\frac{2 x}{3})}{2} d x

ⅰ

\int \frac{1 + \cos (\frac{2 x}{3})}{2} d x

\int \frac{1}{2} d x + \int \frac{\cos (\frac{2 x}{3})}{2} d x

\int \frac{1}{2} d x + \int \frac{\cos (\frac{2 x}{3})}{2} d x

For the math expression:

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

, since (sin(a))' = a'*cos(a), we try differentiating

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

2) Let's differentiate:

\frac{d}{d x} (\sin (\frac{2 x}{3}))

Apply the differentiation rule:

\frac{d}{d x} (\sin (f)) = \frac{d}{d x} (f) \cdot \cos (f)

with f =

\frac{2 x}{3}

ⓘ The derivative of sin(x) is cos(x):

\frac{d}{d x} (\sin (\frac{2 x}{3}))

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

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

ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u'. The derivative of x is 1:

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

\frac{2}{3}

\frac{2}{3} \cdot \cos (\frac{2 x}{3})

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{2}{3} \cdot \cos (\frac{2 x}{3})

\frac{2 \cdot \cos (\frac{2 x}{3})}{3}

\frac{2 \cdot \cos (\frac{2 x}{3})}{3}

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

ⓘ The derivative of

\frac{1}{2} \cdot x

\frac{1}{2}

, so

\frac{1}{2} \cdot x

is an antiderivative of

\frac{1}{2}

\int \frac{1}{2} d x

\frac{1}{2} \cdot x

■The derivative of

\frac{3}{4} \cdot \sin (\frac{2 x}{3})

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

, so

\frac{3}{4} \cdot \sin (\frac{2 x}{3})

is an antiderivative of

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

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

\frac{3}{4} \cdot \sin (\frac{2 x}{3})

\frac{1}{2} \cdot x + \frac{3}{4} \cdot \sin (\frac{2 x}{3})

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{2} \cdot x

\frac{1 \cdot x}{2}

\frac{x}{2}

\frac{3}{4} \cdot \sin (\frac{2 x}{3})

\frac{3 \cdot \sin (\frac{2 x}{3})}{4}

\frac{x}{2} + \frac{3 \cdot \sin (\frac{2 x}{3})}{4}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

\frac{x}{2} + \frac{3 \cdot \sin (\frac{2 x}{3})}{4} + C