The integral e^x ((x-1)/2x^2) dx is equal to: Option 1: e^x/x + C, where C is constant of integration Option 2: e^x/2x + C, where C is constant of integration Option 3: e^x x + C, where C is constant of integration Option 4: x^2 e^x + C, where C is constant of integration

Answer: e^x/2x + C, where C is constant of integration. Solution: Use e^x[f(x) + f'(x)]dx = e^x f(x) + C. Write (x-1)/(2x^2) = 1/(2x) - 1/(2x^2). Take f(x) = 1/(2x), then f'(x) = -1/(2x^2). So the integral equals e^x/(2x) + C.

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

$\frac{e^x}{x} + C$ , where C is constant of integration

$\frac{e^x}{2x} + C$ , where C is constant of integration

$e^x x + C$ , where C is constant of integration

$x^2 e^x + C$ , where C is constant of integration

✅ Correct Option: 2

Use $\int e^x[f(x) + f'(x)]dx = e^x f(x) + C$ . Write $(x-1)/(2x^2) = 1/(2x) - 1/(2x^2)$ . Take $f(x) = 1/(2x)$ , then $f'(x) = -1/(2x^2)$ . So the integral equals $e^x/(2x) + C$ .

2026: 21 May Shift 2

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The value of $\int_{-\pi/2}^{\pi/2}(\sin|x|+\cos|x|)dx$ equals:

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Match the LIST-I with LIST-II

LIST-I (Definite integral)		LIST-II (Value of definite integral)
A. $\displaystyle\int_4^9 \dfrac{1}{\sqrt{x}}\,dx$		I. $\dfrac{2}{3}$
B. $\displaystyle\int_{-1}^1 x^3\,dx$		II. 1
C. $\displaystyle\int_{-1}^1 x^2\,dx$		III. 0
D. $\displaystyle\int_1^2 [x]\,dx$ , where $[\cdot]$ denotes greatest integer function		IV. 2

Choose the correct answer from the options given below:

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The integral ∫ex(x−12x2)dx\int e^x \left(\frac{x-1}{2x^2}\right) dx∫ex(2x2x−1​)dx is equal to:

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