Indefinite Integration
Integral Calculus-1
star_batch_jee_advanced_2025
Grade 12
Question:
If $\int\left(x^{2010} + x^{804} + x^{402}\right)\left(2x^{1008} + 5x^{402} + 10\right)^{10a}dx = \frac{1}{10a}\left(2x^{2010} + 5x^{804} + 10x^{402}\right)^{10a} + c$, where $c$ is constant then $a$ is equal to
Step-by-Step Solution
Key Concept: Identify the substitution $t = 2x^{2010}+5x^{804}+10x^{402}$ such that its derivative exactly matches the remaining factors in the integrand.
Substitute $t = 2x^{2010} + 5x^{804} + 10x^{402}$, so $dt = 4020(x^{2009}+x^{803}+x^{401})dx$. The integral becomes $\int\frac{1}{4020}t^{402}dt = \frac{1}{4020} \cdot \frac{t^{403}}{403} + c = \frac{(2x^{2010}+5x^{804}+10x^{402})^{403}}{4020 \cdot 403} + c$. Note that $a = 403 \Rightarrow a - 400 = 3$.
Correct Answer: 403