Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
If the primitive of the function $f(x) = \frac{x^{2009}}{(1 + x^2)^{1006}}$ w.r.t $x$ is equal to $\frac{1}{n}\left(\frac{x^2}{1 + x^2}\right)^m + C$ then find the value of $(m + n)$ (where $m,n \in \mathbb{N}$)
Step-by-Step Solution
Key Concept: The substitution $1 + x^2 = t$ combined with a secondary substitution $y = 1 - 1/t$ reduces the integral to a simple power form.
For $I = \int \frac{x^{2009}}{(1+x^2)^{1006}}dt$, substitute $1 + x^2 = t$ so $2xdx = dt$. Rewrite as $I = \frac{1}{2}\int\frac{(t-1)^{1004}}{t^{1006}}dt = \frac{1}{2}\int y^{1004}dy$ where $y = 1 - \frac{1}{t}$. This gives $I = \frac{1}{2} \cdot \frac{y^{1005}}{1005} = \frac{x^{1005}}{2 \times 1005} + c$, yielding $m = 1005, n = 2010$, so $m + n = 3015$.
Correct Answer: 3,4