Question

If a continuous function $f$ satisfies ${\int }_0^{x^2}f\left(t\right)dt=x^2(1+x)$ then $f\left(4\right)$ is equal to

• A. $7$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is B $4$

${\int }_0^{x^2}f\left(t\right)dt=x^2(1+x)$

Differentiating w.r.t.$x$,

We know that, $\frac{d}{dx}{\int }_{a\left(x\right)}^{b\left(x\right)}f\left(x\right)dx$$=f\left(b\right(x\left)\right)\frac{d}{dx}b\left(x\right)-f\left(a\right(x\left)\right)\frac{d}{dx}a\left(x\right)$

$2xf\left(x^2\right)=2x+3x^2$

Substitute $x=2$

$2\times 2\times f\left(4\right)=2\times 2+3\times 4$

$\Rightarrow f\left(4\right)=\frac{16}{4}=4$