Indefinite Integration 3 Question 10
10. Let $f:(0, \infty) \rightarrow R$ and $F(x)=\int _0^{x} f(t) d t$.
If $F\left(x^{2}\right)=x^{2}(1+x)$, then $f(4)$ equals
(a) $\frac{5}{4}$
(b) 7
(c) 4
(d) 2
Show Answer
Answer:
Correct Answer: 10. (c)
Solution:
- Given, $F(x)=\int _0^{x} f(t) d t$
By Leibnitz’s rule,
$$ F^{\prime}(x)=f(x) $$
But $\quad F\left(x^{2}\right)=x^{2}(1+x)=x^{2}+x^{3}$
[given]
$$ \begin{array}{ll} \Rightarrow & F(x)=x+x^{3 / 2} \Rightarrow \quad F^{\prime}(x)=1+\frac{3}{2} x^{1 / 2} \\ \Rightarrow & f(x)=F^{\prime}(x)=1+\frac{3}{2} x^{1 / 2} \quad \text { [from Eq. (i)] } \\ \Rightarrow & f(4)=1+\frac{3}{2}(4)^{1 / 2} \Rightarrow f(4)=1+\frac{3}{2} \times 2=4 \end{array} $$
