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Dmitrij [34]
3 years ago
11

Which of the following integrals cannot be evaluated using a simple substitution?

Mathematics
2 answers:
vitfil [10]3 years ago
7 0
The third one, not sure about any other ones.
Burka [1]3 years ago
5 0

The integral \boxed{\int {{x^2}{e^{{x^2}}}dx} } cannot be evaluated by using a simple substitution.Option (c) is correct.

Further explanation:

Given:

The integrals are as follows,

(a). \int {x{e^{{x^2}}}dx}

(b). \int {{e^{\left( {2x + 3} \right)}}dx}

(c). \int {{x^2}{e^{{x^2}}}dx}

(d). None of these.

Explanation:

Option (a)

The integral is I = \int {x{e^{{x^2}}}dx}.

Substitute t for {x^2} in integral I = \int {x{e^{{x^2}}}dx}.

\begin{aligned}{x^2} &= t\\2xdx &= dt\\\end{aligned}

The integral can be obtained as follows,

\begin{aligned}I&= \int {{e^t}dt}\\&= {e^t} + C\\&= {e^{{x^2}}} + C\\\end{aligned}

Option (a) can be evaluated by simple substitution.

Option (b)

The integral is \int {{e^{\left( {2x + 3} \right)}}dx}

The integral can be obtained as follows,

\begin{aligned}I&= \int {{e^{\left( {2x + 3} \right)}}dx}\\&= \frac{1}{2} \times {e^{\left( {2x + 3} \right)}} + C\\\end{aligned}

Option (b) can be evaluated by simple substitution.

Option (c)

The integral is I = \int {{x^2}{e^{{x^2}}}dx}.

Substitute t for {x^2} in integral I = \int {{x^2}{e^{{x^2}}}dx}.

\begin{aligned}{x^2}&= t \\2xdx &= dt\\\end{aligned}

The integral can be obtained as follows,

I = \int {\sqrtt  \times {e^t}dt}

Option (c) cannot be evaluated by simple substitution as we have to use integration by parts.

The integral \boxed{\int {{x^2}{e^{{x^2}}}dx} } cannot be evaluated by using a simple substitution.Option (c) is correct.

Learn more:

  1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
  2. Learn more about equation of circle brainly.com/question/1506955.
  3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Linear equation

Keywords: integral, evaluated, simple substitution, integration, differentiation, exponential.

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