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

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

medium 2 marks Class 12 · Mathematics · Integrals

Answer

\frac{1}{4}

The correct answer is $\frac{1}{4}$ — Hence the value of the integral is $\frac{1}{4}$ .

Recognize that $\int_{0}^{\infty} x e^{- 2 x}, d x$ is of the form $\int_{0}^{\infty} x^{n} e^{- k x}, d x$ which equals $\frac{n !}{k ^{, n + 1}}$ for $n \in N$ .
Here $n = 1$ and $k = 2$ , so the integral equals $\frac{1 !}{2 ^{, 2}} = \frac{1}{4}$ .
Alternatively, substitute $t = 2 x$ ; then $x = t /2$ , $d x = d t /2$ and the integral becomes $\frac{1}{4} \int_{0}^{\infty} t e^{- t}, d t = \frac{1}{4} Γ (2) = \frac{1}{4} \cdot 1 = \frac{1}{4}$ .
Hence the value of the integral is $\frac{1}{4}$ .

Common mistake: Misapplying the gamma function formula or forgetting the correct exponent on

k

Concept tested: Improper integral with exponential decay