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

⚠ 2x is read as the product 2*x: for a power, write 2^x‖1/(sqrt(2x))cos(sqrt(2x)) is read as 1/(sqrt(2x))*cos(sqrt(2x)): use (...)*(...) for multiplication, use (...)^(...) for a power

Before integrating, let's calculate:

\frac{1}{\sqrt{2 x}} \cdot \cos (\sqrt{2 x})

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

Let's compute the integral:

\int(\frac{\cos (\sqrt{2 x})}{\sqrt{2 x}})dx

For the math expression:

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

, we know the derivative formula:

\frac{d}{d x} (\sin (a)) = \cos (a)

, so we try differentiating with sin

Let's perform the differentiation:

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

Apply the differentiation rule:

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

with f =

\sqrt{2 x}

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

Apply the differentiation rule:

\frac{d}{d x} (\sqrt{f}) = \frac{d}{d x} (f) \cdot \frac{1}{2 \sqrt{f}}

with f =

2 x

\frac{d}{d x} (2 x) \cdot \frac{1}{2 \cdot \sqrt{2 x}} \cdot \cos (\sqrt{2 x})

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

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

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

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

\sin (\sqrt{2 x})

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

\sin (\sqrt{2 x}) + C