What is the integral of cos(sqrt(2x)) ?

The integral of cos(sqrt(2x)) is sqrt(2)sqrt(x)sin(sqrt(2)sqrt(x))+cos(sqrt(2)sqrt(x))+C

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integral of cos(sqrt(2x))

\int cos (2 x) d x

2 x sin (2 x) + cos (2 x) + C

Solution steps

\int cos (2 x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int cos (v) \cdot 2 v d v

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

Take the constant out:

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

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

Apply Integration By Parts

: v sin (v) - \int sin (v) d v

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

\int sin (v) d v = - cos (v)

= \frac{1}{2} \cdot 2 (v sin (v) - (- cos (v)))

Substitute back

= \frac{1}{2} \cdot 2 (2 x sin (2 x) - (- cos (2 x)))

Simplify

\frac{1}{2} \cdot 2 (2 x sin (2 x) - (- cos (2 x))) : 2 x sin (2 x) + cos (2 x)

= 2 x sin (2 x) + cos (2 x)

Add a constant to the solution

= 2 x sin (2 x) + cos (2 x) + C

\int (tan (x))^{4} d x

\frac{d}{d x} (\frac{x ^{2} - 5 x + 4}{∣ x - 1 ∣})

\frac{d}{d x} (x^{1 - a})

\frac{\partial}{\partial x} (e^{\frac{y}{x}} \cdot y)

x \to 1 lim (\frac{e ^{x - 1} - 1}{ln ^{2} ( x )})

What is the integral of cos(sqrt(2x)) ?
The integral of cos(sqrt(2x)) is sqrt(2)sqrt(x)sin(sqrt(2)sqrt(x))+cos(sqrt(2)sqrt(x))+C