1.

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. \(\frac 34\)

Answer» 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 = \(\rm \frac {dt}{3}\)

x01
t12

 

Now,

I = \(\rm \frac{1}{3}\int_1^2 tdt\)

= \(\rm \frac{1}{3} \left[\frac{t^2}{2} \right ]_1^2\)

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


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