$\int e^{x}\big[f(x)+f'(x)\big]\,dx$ equals
A$e^{x}f'(x)+C$
B$f(x)+f'(x)+C$
C$e^{x}f(x)+C$
D$e^{x}\big[f(x)-f'(x)\big]+C$
Answer & Solution
Correct answer: C. $e^{x}f(x)+C$
1. Split into $\int e^{x}f(x)\,dx+\int e^{x}f'(x)\,dx$.
2. Integrate $\int e^{x}f(x)\,dx$ by parts: $e^{x}f(x)-\int e^{x}f'(x)\,dx$.
3. Adding back $\int e^{x}f'(x)\,dx$, the two integral terms cancel.
4. Result: $e^{x}f(x)+C$.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.328_
Related questions
$\displaystyle\int_2^3 \dfrac{x\,dx}{x^2+1}$ equals$\displaystyle\int_0^{2a} f(x)\,dx$ equals $2\displaystyle\int_0^{a} f(x)\,dx$ precisely wIf $f(a+b-x)=f(x)$, then $\displaystyle\int_a^b x\,f(x)\,dx$ is equal to$\displaystyle\int \frac{dx}{e^{x}+e^{-x}}$ is equal toA rational function $\dfrac{P(x)}{Q(x)}$ is called proper when$\displaystyle\int_0^{\pi/4} \tan x\,dx$ equals$\displaystyle\int_0^{1} x e^{x^2}\,dx$ equalsThe value of $\displaystyle\int_0^{\pi/2} \log\!\left(\dfrac{4+3 in x}{4+3\cos x}\right)dx