Definite Integration
Grade 12
Question:
<p>If f(x) = x<sup>3</sup> + 3x + 4 then the value of <span class="math-tex">\(\int_\limits{-1}^{1}\)</span> f(x) dx + <span class="math-tex">\(\int_\limits{0}^{4}\)</span> f<sup>-1</sup> (x) dx equals:</p>
<p style="display:inline"><span class="math-tex">\(\frac{21}{4}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{23}{4}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{17}{4}\)</span></p>
<p style="display:inline">4</p>
Step-by-Step Solution
<p><span class="math-tex">\(\int_{-1}^{0} f(x) d x+\int_{0}^{1} f(x) d x+\int_{0}^{4} f^{-1}(x) d x=\int_{0}^{1} f(x) d x\)</span><br />
<span class="math-tex">\(=\left(\frac{x^{4}}{4}+\frac{3 x^{2}}{2}+4 x\right)_{0}^{1}\)</span><br />
<span class="math-tex">\(=\frac{1}{4}+\frac{3}{2}+4=\frac{23}{4}\)</span></p>
Correct Answer: B