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.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Linear equation
Keywords: integral, evaluated, simple substitution, integration, differentiation, exponential.
