Evaluate each of the following improper integrals. a) \ \int_{-\infty}^0 7e^{-7x+2}dx \\ b) \...

Question:

Evaluate the following improper integral.

{eq}\int^\infty_0 7e^{-7x+2}dx {/eq}

U - Substitution:

Not every integral needs a substitution, but the integrals that is difficult to integrate directly needs substitution in a special way.

If the integral is in or can made in the form,

{eq}\displaystyle \int f(g(x)) \cdot g'(x) \ dx {/eq} ,

then substitute {eq}u = g(x) \Rightarrow du = g'(x) \ dx {/eq}

{eq}\displaystyle \int f(u) \ du {/eq}

Answer and Explanation:

Let {eq}\displaystyle I = \int^{\infty}_0 7e^{-7x+2}dx {/eq}


Use the substitution method,

Let {eq}\displaystyle u = - 7x + 2 \Rightarrow du = -7\ dx \Rightarrow 7\ dx = -du {/eq}


Then,

{eq}\displaystyle \begin{align*} \Rightarrow I &= \int^{\infty}_0 e^u (-du) \\ \Rightarrow I &= - \int^{\infty}_0 e^u du \\ \Rightarrow I &= - \Biggr [ e^u \Biggr ]^{\infty}_0 \\ \end{align*} {/eq}


Now substitute back the value of {eq}u = -7x + 2 {/eq},

{eq}\Rightarrow I = - \Biggr [ e^{ -7x + 2} \Biggr ]^{\infty}_0 \\ \Rightarrow I = - (0 - e^2) \\ \Rightarrow I = e^2 {/eq}


Learn more about this topic:

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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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