Home -> Solved problems -> Calculate the integral \int_{0}^{1}(-1)^{x} dx

## Calculate the integral $$\int_{0}^{1}(-1)^{x} d x$$ ### Solution

What is the integral from 0 to 1 of negative 1 raised to the power of $$x$$
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\begin{aligned} e^{i \theta} &=\cos \theta+i \sin \theta \\\\ e^{i \pi} &=\cos \pi+i \sin \pi \\\\ &=-1 \end{aligned}
\begin{aligned} (-1)^{x}&=\left(e^{i \pi}\right)^{x} \\\\ (-1)^{x}&=e^{i \pi x} \\\\ \text { for } &x>0\\\\ \int_{0}^{1}(-1)^{x} d x &=\int_{0}^{1} e^{i \pi x} d x \\\\ &=\left(\frac{e^{i \pi x}}{i \pi}\right)_{0}^{1} \\\\ &=\frac{e^{i \pi}}{i \pi}-\frac{1}{i \pi}\\\\ &=\frac{-1}{i \pi}-\frac{1}{i \pi} \\\\ &=\frac{-2}{i \pi} \end{aligned}
Home -> Solved problems -> Calculate the integral \int_{0}^{1}(-1)^{x} dx

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Calculate the integral $$\int_{0}^{1}(-1)^{x} d x$$
Home -> Solved problems -> Calculate the integral \int_{0}^{1}(-1)^{x} dx

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