Evaluate the improper integral $d i s pl a y s t y l e in t_{0}^{in f t y} x^{4} e^{- 2 x}, d x$ using the gamma function or standard formulas.

hard 4 marks Class 12 · Mathematics · Integrals

Answer

\frac{3}{4}

The correct answer is $\frac{3}{4}$ — Hence the value of the integral is $\frac{3}{4}$ .

Step-by-step solution

Recognise the integral as of the form $in t_{0}^{in f t y} x^{n} e^{- a x}, d x$ .
Use the formula $\int_{0}^{\infty} x^{n} e^{- a x}, d x = \frac{n !}{a ^{, n + 1}}$ for integer $n g e 0$ and $a > 0$ .
Here $n = 4$ and $a = 2$ .
Compute $4! = 24$ and $2^{, 5} = 32$ .
Divide: $\frac{24}{32} = \frac{3}{4}$ .
Hence the value of the integral is $\frac{3}{4}$ .

Common mistake: Incorrect exponent in denominator of the formula

Concept tested: Gamma function and exponential integrals

Published 10 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked