Correct Answer - Option 3 :
\(\frac 12\)
Concept:
\(\rm \int x^n dx = \frac{x^{n+1}}{n+1}+c\)
Calculation:
I = \(\rm \int_0^1 x^2(1+x^3)dx\)
Let 1 + x3 = t
Differentiating with respect to x, we get
⇒ (0 + 3x2)dx = dt
⇒ x2 dx = \(\rm \frac {dt}{3}\)
Now,
I = \(\rm \frac{1}{3}\int_1^2 tdt\)
= \(\rm \frac{1}{3} \left[\frac{t^2}{2} \right ]_1^2\)
= \(\rm \frac{1}{6} [4-1] = \frac{3}{6} = \frac{1}{2}\)