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

The integral [tex]\boxed{\int {{x^2}{e^{{x^2}}}dx} }[/tex] cannot be evaluated by using a simple substitution.Option (c) is correct.
Further explanation:
Given:
The integrals are as follows,
(a). [tex]\int {x{e^{{x^2}}}dx}[/tex]
(b). [tex]\int {{e^{\left( {2x + 3} \right)}}dx}[/tex]
(c). [tex]\int {{x^2}{e^{{x^2}}}dx}[/tex]
(d). None of these.
Explanation:
Option (a)
The integral is [tex]I = \int {x{e^{{x^2}}}dx}.[/tex]
Substitute [tex]t[/tex] for [tex]{x^2}[/tex] in integral [tex]I = \int {x{e^{{x^2}}}dx}.[/tex]
[tex]\begin{aligned}{x^2} &= t\\2xdx &= dt\\\end{aligned}[/tex]
The integral can be obtained as follows,
[tex]\begin{aligned}I&= \int {{e^t}dt}\\&= {e^t} + C\\&= {e^{{x^2}}} + C\\\end{aligned}[/tex]
Option (a) can be evaluated by simple substitution.
Option (b)
The integral is [tex]\int {{e^{\left( {2x + 3} \right)}}dx}[/tex]
The integral can be obtained as follows,
[tex]\begin{aligned}I&= \int {{e^{\left( {2x + 3} \right)}}dx}\\&= \frac{1}{2} \times {e^{\left( {2x + 3} \right)}} + C\\\end{aligned}[/tex]
Option (b) can be evaluated by simple substitution.
Option (c)
The integral is [tex]I = \int {{x^2}{e^{{x^2}}}dx}.[/tex]
Substitute [tex]t[/tex] for [tex]{x^2}[/tex] in integral [tex]I = \int {{x^2}{e^{{x^2}}}dx}.[/tex]
[tex]\begin{aligned}{x^2}&= t \\2xdx &= dt\\\end{aligned}[/tex]
The integral can be obtained as follows,
[tex]I = \int {\sqrtt \times {e^t}dt}[/tex]
Option (c) cannot be evaluated by simple substitution as we have to use integration by parts.
The integral [tex]\boxed{\int {{x^2}{e^{{x^2}}}dx} }[/tex] cannot be evaluated by using a simple substitution.Option (c) is correct.
Learn more:
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Linear equation
Keywords: integral, evaluated, simple substitution, integration, differentiation, exponential.