Animated Solution for Mathematics - Definite Integration: The value of ${\int }_0^{1}4x^3\left\{\frac{d^2}{d{x}^2}(1-x^2{)}^5\right\}dx$ is

Imagine you are standing before a complex integral:

At first glance, it looks like a monster. You see a polynomial, $4x^3$, locked in a dance with a second-order derivative.

In the world of JEE Advanced, we do not brute force; we strategize. This integral is a classic case for Integration by Parts (IBP). The core idea is to 'absorb' the derivative to reduce its order.

We choose our tools carefully. Let $u=4x^3$. Differentiating $u$ gives us $du=12x^{2}dx$, which is a simpler polynomial.

Now, for the $dv$ part, we take the entire derivative term:

When we integrate $dv$, we get:

We now apply the IBP formula: $\int udv=[uv]-\int vdu$.

Let us evaluate the boundary term $[uv{]}_0^1$. We have:

At $x=1$, the term $(1-x^2{)}^4$ inside the derivative becomes zero. At $x=0$, the $4x^3$ term becomes zero. Consequently, the entire boundary term vanishes.

We are left with the integral $-{\int }_0^{1}vdu$. Substituting $v=-10x(1-x^2{)}^4$ and $du=12x^{2}dx$, the integral becomes:

Now, we have a standard integral. Let $t=1-x^2$. Then $dt=-2xdx$, or $xdx=-\frac{1}{2}dt$.

Since $x^2=1-t$, the limits change from $x=0\to t=1$ and $x=1\to t=0$. The integral transforms into:

Integrating term by term, we get:

Plugging in the limits:

The monster is tamed, and the final answer is 2.