PYQ Vault
EbooksTutor

Integral of e^x (f(x) + f'(x)) recognised as e^x f(x)

This type of question, asked 3 times: 27 Jun 2022, 22 Jan 2025, 23 Jan 2026 S1. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    27 Jun 2022

    69
    ∫(x2+1)ex(x+1)2dx=f(x)ex+C\int\frac{\left( x^{2}+ 1 \right)e^{x}}{(x + 1)^{2}}dx = f(x)e^{x}+ C, Where CC is a constant, then d3fdx3\frac{d^{3}f}{dx^{3}} at x=1x = 1 is equal to:

    Tap an option to check your answer.

  2. Question 2

    22 Jan 2025

    136
    If ∫ex(xsin⁡−1x1−x2+sin⁡−1x(1−x2)3/2+x1−x2)dx=g(x)+C\int e^{x}\left(\frac{x\sin^{-1}x}{\sqrt{1-x^{2}}}+\frac{\sin^{-1}x}{\left(1-x^{2}\right)^{3/2}}+\frac{x}{1-x^{2}}\right)dx = g(x)+C, where CC is the constant of integration, then g(12)g\left(\frac{1}{2}\right) equals:

    Tap an option to check your answer.

  3. Question 3

    23 Jan 2026 S1

    52
    Let f(x)=∫ (2−x2)⋅ex(1+x)(1−x)3/2dxf(x) =\int_{}^{}\ \frac{\left( 2 -x^{2} \right)\cdot e^{x}}{(\sqrt{1 + x})(1 - x)^{3/2}}dx. If f(0)=0f(0) = 0, then f(12)f\left( \frac{1}{2} \right) is equal to :

    Tap an option to check your answer.