Test Index

MHTCET 13 Apr 2026 Shift 2 Paper

Question : 127 of 150

Marks: +1, -0

\int x^{2} \cdot e^{x} d x=e^{x} f(x)+c

f(x)

is ...

Solution:

Concept:

Integrate by parts to find

f(x)

, then complete the square to find its minimum value.

Explanation:

Given:

\int x^{2} e^{x}\, dx = e^{x}f(x) + c

Using the standard formula

\int x^{2}e^{x}\,dx = e^{x}(x^{2} - 2x + 2) + c

we get:

f(x) = x^{2} - 2x + 2

Rewrite by completing the square:

f(x) = (x^{2} - 2x + 1) + 1 = (x - 1)^{2} + 1

Since

(x - 1)^{2} \geq 0

, the minimum value of

f(x)

0 + 1 = 1

Answer:

The minimum value is

1

Correct option: C.

Go to Question:

Prev Question

Next Question