What is the integral of sqrt(2[1-cos^2(\sqrt{2)x)]} ?

The integral of sqrt(2[1-cos^2(\sqrt{2)x)]} is -cos(sqrt(2)x)+C

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integral of sqrt(2[1-cos^2(\sqrt{2)x)]}

\int 2 [1 - cos^{2} (2 x)] d x

- cos (2 x) + C

Solution steps

\int 2 [1 - cos^{2} (2 x)] d x

Simplify

2 [1 - cos^{2} (2 x)] : 2 sin (2 x)

= \int 2 sin (2 x) d x

Take the constant out:

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

= 2 \cdot \int sin (2 x) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= 2 \frac{1}{2} (- cos (u))

Substitute back

u = 2 x

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

Simplify

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

= - cos (2 x)

Add a constant to the solution

= - cos (2 x) + C

\int cos (x) (2 + sin (x))^{6} d x

\frac{d y}{d t} + t y^{2} = 0

\frac{4 x}{e ^{x}}

\frac{d}{d x} (\frac{1 + 1}{x})

1 - x^{2}, ∣ x ∣

What is the integral of sqrt(2[1-cos^2(\sqrt{2)x)]} ?
The integral of sqrt(2[1-cos^2(\sqrt{2)x)]} is -cos(sqrt(2)x)+C