For the following problem find a substitution w and a constant k so that the integral has the...


For the following problems find a substitution {eq}w {/eq} and a constant {eq}k {/eq} so that the integral has the form {eq}\int ke^w dw. {/eq}

a. {eq}\int xe ^{-x^2} dx {/eq}

b. {eq}\int \sqrt{e^r dr} {/eq}

c. {eq}\int e^{2t}e^{3t-4} dt {/eq}

Indefinite Integral in Calculus:

Integration is one of the major concept in mathematics.

We are given an indefinite integral and we'll solve this problem by using u-substitution which is a process of temporary substitution of the given variable to get the standard form of the integrand to make the integration process easier.

Answer and Explanation:


We are given: {eq}\displaystyle \int xe ^{-x^2} dx {/eq}

Apply u-substitution {eq}w= -x^2 dx \rightarrow \ dw = -2x \ dx {/eq}

{eq}= \displaystyle \int -\dfrac{1}{2}e ^{w} dx {/eq}

It is in the form {eq}\displaystyle \int ke^w dw {/eq}. Where {eq}k = -\dfrac{1}{2} {/eq}.


We are given: {eq}\displaystyle \int \sqrt{e^r} \ dr {/eq}

Apply the rule {eq}\displaystyle \sqrt{ a^m}= a^{m/2} {/eq}

{eq}= \displaystyle \int e^{r/2 }dr {/eq}

Apply u-substitution {eq}w= \dfrac{r}{2} \rightarrow \ dw =\dfrac{1}{2}\ dr {/eq}

{eq}= \displaystyle \int 2 e ^{w} dx {/eq}

It is in the form {eq}\displaystyle \int ke^w dw {/eq}. Where {eq}k = 2 {/eq}.


We are given: {eq}\displaystyle \int e^{2t}e^{3t-4} dt {/eq}

Apply the rule {eq}\displaystyle e^a e^b = e^{a+b} {/eq}

{eq}= \displaystyle \int e^{2t +3t-4}dt {/eq}

{eq}= \displaystyle \int e^{5t-4}dt {/eq}

Apply u-substitution {eq}w=5t -4 \rightarrow \ dw =5 \ dt {/eq}

{eq}= \displaystyle \int \dfrac{1}{5}e ^{w} dx {/eq}

It is in the form {eq}\displaystyle \int ke^w dw {/eq}. Where {eq}k = \dfrac{1}{5} {/eq}.

Learn more about this topic:

How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5

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