Question

What is the integral of ${{e}^{0.5x}}?$

Answer 1

Hint: We use the basic integration formula to solve this question. We need to know the standard formula for the integration of an exponent which is given as, $\int{{{e}^{x}}}dx={{e}^{x}}+c.$ Here, c represents a constant value which is known as the constant of integration. Since the power of e is not just x, we use the substitution method where we substitute a variable $t=0.5x.$ Simplifying for this new variable and substituting the value of t after obtaining the final expression, we get the answer.

Complete step-by-step solution:
We solve this question by using the basic integration formula.
$\Rightarrow \int{{{e}^{0.5x}}dx}\ldots \ldots \left( 1 \right)$
To solve this expression, we need to use the substitution method. Let us substitute the power of the exponential term as,
$\Rightarrow t=0.5x$
Now, we differentiate both sides of the equation as,
$\Rightarrow dt=0.5dx$
Rearranging the terms, we get the value of dx as,
$\Rightarrow \dfrac{dt}{0.5}=dx$
We know the value of 0.5 in terms of fractions is given by $\dfrac{1}{2}.$ Substituting this,
$\Rightarrow \dfrac{dt}{\dfrac{1}{2}}=dx$
$\Rightarrow 2dt=dx$
Using this value of dx in equation 1 along with the new variable t in place of 0.5x,
$\Rightarrow \int{{{e}^{t}}.2dt}$
Taking the constant outside the integral,
$\Rightarrow 2\int{{{e}^{t}}dt}$
We now apply the formula for the integral of an exponential given as $\int{{{e}^{x}}}dx={{e}^{x}}+c.$
\[\Rightarrow 2.\left( {{e}^{t}}+c \right)\]
Now, we need to substitute the value of t in terms of x to obtain the solution.
\[\Rightarrow 2.\left( {{e}^{0.5x}}+c \right)\]
We multiply both the terms by 2 and we know that the constant term multiplied by 2 is another constant term ${{c}_{1}}$ such that ${{c}_{1}}=2c.$
\[\Rightarrow 2{{e}^{0.5x}}+2c=2{{e}^{0.5x}}+{{c}_{1}}\]
Hence, the integral of ${{e}^{0.5x}}$ is \[2{{e}^{0.5x}}+{{c}_{1}}.\]

Note: It is important to know the basic formula of integration in order to solve such problems. We can also solve this question by directly applying the formula $\int{{{e}^{nx}}}dx=\dfrac{{{e}^{nx}}}{n}+c.$ Here, the n value is the coefficient of x which is 0.5 in this case.