Calculus

Integrals

Medium

common

The value of $\int_0^1 x e^x dx$ is:

$\frac{1}{e}$

$e$

$1$

$-1$

✅ Correct Option: 3

When integrating a product of different types of functions like $x$ and $e^x$ , use integration by parts.

$\int u \, dv = uv - \int v \, du$

Using the ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential), choose the function that appears first as $u$ .

Let $u = x$ (Algebraic) and $dv = e^x dx$ (Exponential)

Then $du = dx$ and $v = e^x$

Applying integration by parts:

$\int x e^x dx = x \cdot e^x - \int e^x \cdot dx$

$= x e^x - e^x + C$

$= e^x(x - 1) + C$

Evaluating the definite integral from 0 to 1:

$\int_0^1 x e^x dx = \left[e^x(x - 1)\right]_0^1$

At $x = 1$ :

$e^1(1 - 1) = 0$

At $x = 0$ :

$e^0(0 - 1) = -1$

$\int_0^1 x e^x dx = 0 - (-1)$

$= 1$

Therefore, the value is $1$ .

2025: 3 June Shift 2

If $\int \sqrt\frac{1-x}{{1+x}} dx = a\sqrt{1-x^2} + \beta \sin^{-1}x + C$ , Where C is an arbitrary constant, then which of the following are TRUE?

(A) $\alpha = 1$

(B) $\alpha = -1$

(D) $\beta = -1$

Choose the correct answer from the options given below:

2025: 13 May Shift 1

The value of which of the following integrals is zero?

(A) $\int_0^1 x dx$

(B) $\int_{-1}^1 x dx$

(D) $\int_0^1 \log\left(\frac{x}{1-x}\right) dx$

Choose the correct answer from the options given below:

2025: 29 May Shift 2

$\int \frac{dx}{(1+5\sin^2 x)}$ is equal to

2025: 27 May Shift 1

$\int \frac{dx}{e^x + e^{-x}}$ is equal to

2023: 30 May Shift 3

$\int \frac{\sqrt{\tan x}}{\sin x \cos x} dx$ equals :

2025: 30 May Shift 2

$\int (x^4 + x^2 + 1)d(x^2)$ is equal to: (where c is an integration constant)

The value of ∫01xexdx\int_0^1 x e^x dx∫01​xexdx is:

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The value of $\int_0^1 x e^x dx$ is: