What is the integral of (-1)^{2x} ?

The integral of (-1)^{2x} is 1/(2ln(-1))+C

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integral of (-1)^{2x}

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

\frac{1}{2 ln ( - 1 )} + C

Solution steps

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

Apply u-substitution

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

= \int (- 1)^{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 (- 1)^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{1}{2} \cdot \frac{( - 1 ) ^{u}}{ln ( - 1 )}

Substitute back

u = 2 x

= \frac{1}{2} \cdot \frac{( - 1 ) ^{2 x}}{ln ( - 1 )}

Simplify

\frac{1}{2} \cdot \frac{( - 1 ) ^{2 x}}{ln ( - 1 )} : \frac{1}{2 ln ( - 1 )}

= \frac{1}{2 ln ( - 1 )}

Add a constant to the solution

= \frac{1}{2 ln ( - 1 )} + C

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