Question

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

Answer 1

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.