_0^1 x^2 e^x^3 dx is equal to : Option 1: 1/3(e - 1) Option 2: 2/3(e - 1) Option 3: 3/4(e - 1) Option 4: 3/7(e - 1)

Answer: 1/3(e - 1). Solution: Let u = x^3, then du = 3x^2 dx, so x^2 dx = du/3. When x = 0, u = 0; when x = 1, u = 1. _0^1 x^2 e^x^3 dx = 1/3 _0^1 e^u du = 1/3[e^u]_0^1 = 1/3(e - 1).

Calculus

Integrals

Easy

$\int_{0}^{1} x^2 e^{x^3} \, dx$ is equal to :

$\frac{1}{3}(e - 1)$

$\frac{2}{3}(e - 1)$

$\frac{3}{4}(e - 1)$

$\frac{3}{7}(e - 1)$

✅ Correct Option: 1

Let $u = x^3$ , then $du = 3x^2 \, dx$ , so $x^2 \, dx = \frac{du}{3}$ .

When $x = 0$ , $u = 0$ ; when $x = 1$ , $u = 1$ .

$\int_{0}^{1} x^2 e^{x^3} \, dx = \frac{1}{3} \int_{0}^{1} e^u \, du = \frac{1}{3}[e^u]_{0}^{1} = \frac{1}{3}(e - 1)$ .

2026: 30 May Shift 2

If $0 < x < \frac{\pi}{4}$ , then $\int\sqrt{1-\sin2x}\,dx$ is:

[where C is an arbitrary constant]

2025: 21 May Shift 1

Match List-I with List-II

List-I	List-II
Definite integral	Value
---	---
(A) $\int_{1}^{e} \frac{\log x}{x} dx$	(I) 4
(B) $\int_{-2}^{2} x^3(1 - x^2) dx$	(II) $\frac{1}{2}$
(C) $\int_{1}^{2} x \, dx$	(III) 0
(D) $\int_{-2}^{2} \vert x\vert dx$	(IV) $\frac{3}{2}$

Choose the correct answer from the options given below:

2026: 30 May Shift 1

$\int\frac{\tan^5\sqrt{x}.\sec^2\sqrt{x}}{\sqrt{x}}dx$ is equal to:

2025: 27 May Shift 1

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

2022: 30 Aug Shift 1

$\int x\sqrt{x + 2} \, dx$ is equal to :

2025: 19 May Shift 1

Match List-I with List-II

List-I	List-II
(A) $\int_{-a}^a f(x) dx = 0$	(I) 0
(B) $\int_0^{2a} f(x) dx = 2\int_0^a f(x) dx$	(II) 1
(C) $\int_{-\pi}^{\pi} \cos x dx$	(III) $f$ is an odd function
(D) $\int_{-1}^1 x^{101} dx + 1$	(IV) $f(2a-x) = f(x)$

Choose the correct answer from the options given below:

∫01x2ex3 dx\int_{0}^{1} x^2 e^{x^3} \, dx∫01​x2ex3dx is equal to :

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

$\int_{0}^{1} x^2 e^{x^3} \, dx$ is equal to :