Evaluate the improper integral ∫_0^(∞)x^2e^-3x dx using integration by parts or the gamma function.

Evaluate the improper integral $\int_{0}^{\infty} x^{2} e^{- 3 x} d x$ using integration by parts or the gamma function.

hard 3 marks Class 12 · Mathematics · Integrals

Answer

\frac{2}{27}

The correct answer is $\frac{2}{27}$ — Therefore $\int_{0}^{\infty} x^{2} e^{- 3 x} d x = \frac{2}{27}$ .

Recall the standard result $\int_{0}^{\infty} x^{n} e^{- a x} d x = \frac{n !}{a ^{, n + 1}}$ for $a > 0$ .
Identify $n = 2$ and $a = 3$ .
Compute $2! = 2$ and $3^{, 3} = 27$ .
Therefore $\int_{0}^{\infty} x^{2} e^{- 3 x} d x = \frac{2}{27}$ .

Common mistake: Incorrectly applying the integration by parts formula or forgetting the factor

1/ a

after each integration

Concept tested: Improper integrals with exponential decay

Published 7 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked