e^x (1/x - 2/x^3) dx = Option 1: e^x ((x+1)/x^2) + C Option 2: e^x ((x-1)/x^2) + C Option 3: e^x ((x-2)/x^2) + C Option 4: e^x ((x^2-2)/x^3) + C

Answer: e^x ((x+1)/x^2) + C. Solution: Use e^x[f(x) + f'(x)]dx = e^x f(x) + C. Let f(x) = (x+1)/x^2 = 1/x + 1/x^2. Then f'(x) = -1/x^2 - 2/x^3. So f(x) + f'(x) = 1/x - 2/x^3. Hence the integral = e^x ((x+1)/x^2) + C.

Calculus

Integrals

Medium

$\int e^x \left(\frac{1}{x} - \frac{2}{x^3}\right) dx =$

$e^x \left(\frac{x+1}{x^2}\right) + C$

$e^x \left(\frac{x-1}{x^2}\right) + C$

$e^x \left(\frac{x-2}{x^2}\right) + C$

$e^x \left(\frac{x^2-2}{x^3}\right) + C$

✅ Correct Option: 1

Use $\int e^x[f(x) + f'(x)]dx = e^x f(x) + C$ .

Let $f(x) = \frac{x+1}{x^2} = \frac{1}{x} + \frac{1}{x^2}$ .

Then $f'(x) = -\frac{1}{x^2} - \frac{2}{x^3}$ .

So $f(x) + f'(x) = \frac{1}{x} - \frac{2}{x^3}$ .

Hence the integral $= e^x \left(\frac{x+1}{x^2}\right) + C$ .

2025: 14 May Shift 1

The integral I = $\int \frac{e^{5\log_e x} - e^{4\log_e x}}{e^{3\log_e x} - e^{2\log_e x}} dx$ is equal to

2026: 22 May Shift 1

Match List-I with List-II

Where C is arbitrary constant.

List-I	List-II
(A) $\displaystyle\int \dfrac{dx}{x^2 - a^2} =$	(I) $\log\left\vert x + \sqrt{x^2+a^2}\right\vert + C$
(B) $\displaystyle\int \dfrac{dx}{a^2 - x^2} =$	(II) $\log\left\vert x + \sqrt{x^2-a^2}\right\vert + C$
(C) $\displaystyle\int \dfrac{dx}{\sqrt{x^2 - a^2}} =$	(III) $\dfrac{1}{2a}\log\left\vert \dfrac{x-a}{x+a}\right\vert + C$
(D) $\displaystyle\int \dfrac{dx}{\sqrt{x^2 + a^2}} =$	(IV) $\dfrac{1}{2a}\log\left\vert \dfrac{a+x}{a-x}\right\vert + C$

Choose the correct answer from the options given below:

2026: 19 May Shift 2

$\int 5^x dx$ is

(where C is an arbitrary constant)

2026: 18 May Shift 1

$\int \frac{1}{x^2+4x+8}dx$ is equal to (Where C is an arbitrary constant)

2025: 14 May Shift 1

$\int_1^4 |x - 2|dx$ is equal to

2025: 14 May Shift 1

The integral I = $\int e^x\left(\frac{x - 1}{3x^2}\right) dx$ is equal to

∫ex(1x−2x3)dx=\int e^x \left(\frac{1}{x} - \frac{2}{x^3}\right) dx =∫ex(x1​−x32​)dx=

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

$\int e^x \left(\frac{1}{x} - \frac{2}{x^3}\right) dx =$