Evaluate the integral: ∫3x22x3−1dx a.x3−1+C b.x3+C c.1x3−1+C d.None of the above

Solved: "Evaluate the integral: ∫3x22x3−1dx a.x3−1+C b.x3+C c.1x3−1+C d.None..."

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Caelan

Answered question

2021-10-13

Evaluate the integral:
$\int \frac{3 x^{2}}{2 \sqrt{x^{3} - 1}} d x$
a. $\sqrt{x^{3} - 1} + C$
b. $\sqrt{x^{3}} + C$
c. $\frac{1}{\sqrt{x^{3} - 1}} + C$
d.None of the above

Answer & Explanation

Aniqa O'Neill

Skilled2021-10-14Added 100 answers

Step 1
Consider the integrals,

\int \frac{3 x^{2}}{2 \sqrt{x^{3} - 1}} d x

...(1)
Step 2
Let,

x^{3} - 1 = t^{2} \Rightarrow t = \sqrt{x^{3} - 1}

Differentiating both sides,

3 x^{2} d x - 0 = 2 t d t

3 x^{2} d x = 2 t d t

Step 3
Substitute all value in equation (1) then,

\int \frac{3 x^{2}}{2 \sqrt{x^{3} - 1}} d x = \frac{1}{2} \int \frac{2 t d t}{t}

= \frac{2}{2} \int d t

= \int d t

=t+C
Back substitution,

= \sqrt{x^{3} - 1} + C

Hence first option is correct (a).

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