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Exercise

$\int_0^1z^xdx$

Step-by-step Solution

1

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

$\left[\frac{z^x}{\ln\left|z\right|}\right]_{0}^{1}$
2

Replace the integral's limit by a finite value

$\lim_{c\to1}\left(\left[\frac{z^x}{\ln\left|z\right|}\right]_{0}^{c}\right)$
3

Evaluate the definite integral

$\lim_{c\to1}\left(\frac{z^c}{\ln\left|z\right|}-\frac{z^0}{\ln\left|z\right|}\right)$
4

Simplify the expression

$\lim_{c\to1}\left(\frac{z^c-1}{\ln\left(z\right)}\right)$
5

Evaluate the resulting limits of the integral

$\frac{z-1}{\ln\left(z\right)}$

Final answer to the exercise

$\frac{z-1}{\ln\left(z\right)}$

Try other ways to solve this exercise

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  • Integrate by partial fractions
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  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
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