Find the value of $\rm \int_0^1 x^2(1+x^3)dx$

15 views
in Calculus
closed
Find the value of $\rm \int_0^1 x^2(1+x^3)dx$
1. $\frac 1 4$
2. $\frac 32$
3. $\frac 12$
4. $\frac 34$
5. None of these

by (30.0k points)
selected

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 = $\rm \frac {dt}{3}$

 x 0 1 t 1 2

Now,

I = $\rm \frac{1}{3}\int_1^2 tdt$

$\rm \frac{1}{3} \left[\frac{t^2}{2} \right ]_1^2$

$\rm \frac{1}{6} [4-1] = \frac{3}{6} = \frac{1}{2}$