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

The integral of sqrt(cos^2(x)-sin^2(x)) is \E(2\vert (2x)/2)+C

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integral of sqrt(cos^2(x)-sin^2(x))

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

E (2∣ \frac{2 x}{2}) + C

Solution steps

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

Rewrite using trig identities

= \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

Use the nonelementary integral:

\int cos (x) d x = 2 E (\frac{x}{2} ∣ 2)

= \frac{1}{2} \cdot 2 E (\frac{u}{2} ∣ 2)

Substitute back

u = 2 x

= \frac{1}{2} \cdot 2 E (\frac{2 x}{2} ∣ 2)

Simplify

\frac{1}{2} \cdot 2 E (\frac{2 x}{2} ∣ 2) : E (2∣ \frac{2 x}{2})

= E (2∣ \frac{2 x}{2})

Add a constant to the solution

= E (2∣ \frac{2 x}{2}) + C

(t - 3)^{4}

\frac{d}{d x} (e^{2 x} + 5)

(2 x y^{2} - 7) d x + (2 x^{2} y + 5) d y = 0

\frac{d}{d x} (8 e^{x} + \frac{4}{3 x})

f (x) = x, (25, 5)

What is the integral of sqrt(cos^2(x)-sin^2(x)) ?
The integral of sqrt(cos^2(x)-sin^2(x)) is \E(2\vert (2x)/2)+C