Definite Integration
Grade 12
Question:
<p>Evaluate<span class="math-tex">\(\int_2^3 2 x^2 e^{x^3} d x\)</span>.<br />
</p>
<p style="display:inline"><span class="math-tex">\(\frac{2}{3}\left(e^8-e^{27}\right)\)</span></p>
<p style="display:inline"><span class="math-tex">\(e^{27}-e^8\)</span><br/>
</p>
<p style="display:inline"><span class="math-tex">\(\frac{2}{3}\left(e^{27}-e^8\right)\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{2}{3}\left(e^{27}+e^8\right)\)</span></p>
Step-by-Step Solution
Key Concept: Use the method of substitution by setting $t = x^3$ to simplify the integrand into a basic exponential form.
<p><span class="math-tex">${I}=\int_2^3 2 x^2 e^{x^3} d x$</span><br />
Let <span class="math-tex">${x}^3={t}$</span><br />
Differentiating w.r.t x , we get<br />
<span class="math-tex">$ 3 {x}^2 {dx}={dt} $</span><br />
<span class="math-tex">$ {x}^2 {dx}=\frac{d t}{3} $</span><br />
The new limits<br />
When <span class="math-tex">${x}=2, {t}=8$</span><br />
When <span class="math-tex">${x}=3, {t}=27$</span></p>
<p><span class="math-tex">$ \therefore \int_2^3 2 x^2 e^{x^3} d x=\frac{2}{3} \int_8^{27} e^t d t $</span><br />
<span class="math-tex">$ =\frac{2}{3}\left[e^t\right]_8^{27}=\frac{2}{3}\left(e^{27}-e^8\right) . $</span></p>
Correct Answer: C