Evaluate the integral: integral_{-1}^1 e^{u + 1} du.


Evaluate the integral:

{eq}\displaystyle \int_{-1}^1 e^{u + 1}\ du {/eq}.

Exponents Rule for Integrals:

To simplify the power of the exponential function, we have the general mathematical expression shown below:

{eq}\displaystyle e^{m+n}=e^{m}\cdot e^{n} {/eq}

After that, we'll take out the constant term and simplify obtained exponential expression by the most general rule.

{eq}\int e^x\ dx=e^x+C {/eq}, where,

  • C is the constant of integration.

Answer and Explanation:


The exponential integral expression is:

{eq}\displaystyle I= \int_{-1}^1 e^{u + 1}\ du {/eq}

Using the exponential power rule of exponentials in the above definite integral expression, we get:

{eq}\begin{align*} \displaystyle I&= \displaystyle \int_{-1}^1 e^{u + 1}\ du\\ \displaystyle I&= \displaystyle \int_{-1}^1 e^{u}\cdot e^1\ du\\ \displaystyle I&= \displaystyle e\int_{-1}^1 e^{u}\ du\\ \end{align*} {/eq}

Solving the above definite integral expression, we get:

{eq}\begin{align*} \displaystyle I&= \displaystyle e\left [e^{u} \right ]_{-1}^1\\ &= \displaystyle e\left (e^{1}-e^{-1} \right )\\ &= \displaystyle e\left (e-\frac{1}{e^{1}} \right )\\ &= \displaystyle e\left (\frac{e^2-1}{e} \right )\\ &= \boxed{\displaystyle e^2-1}\\ \end{align*} {/eq}

Learn more about this topic:

Evaluating Definite Integrals Using the Fundamental Theorem

from AP Calculus AB: Exam Prep

Chapter 16 / Lesson 2

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