Question
\(\int \frac{x^{2}}{e^{x^{3}}}dx\)
(A) \(-\frac{1}{3}lne^{x^3}+c\) (B) \(-\frac{1}{3}e^{x^3}+c\) (C) \(-\frac{1}{3e^{x^3}}+c\) (D) \(\frac{1}{3}lne^{x^3}+c\) (E) \(\frac{1}{3e^{x^3}}+c\)
▶️Answer/Explanation
Ans:C
\(\int \frac{x^2}{e^{x^{3}}}dx=-\frac{1}{3}\int e^{-x^{3}}(-3x^{2}dx)=-\frac{1}{3}e^{-x^{3}}+C=-\frac{1}{3e^{-x^{3}}}+C\)
Question
\(\int (x^{3}-3x)dx\)=
(A)\(3x^{2}-3+C\) (B)\(4x^{2}-6x^{2}+C\) (C)\(\frac{x^{4r}}{3}-3x^{2}+C\) (D)\(\frac{x^{4}}{4}-3x+C\) (E)\(\frac{x^{4}}{4}-\frac{3x^{2}}{2}+C\)
▶️Answer/Explanation
Ans:E
\(\int (x^{3}-3x)dx=\frac{1}{4}x^{4}-\frac{3}{2}x^{2}+C\)
Question
An antiderivative for \( \frac{1}{x^{2}-2x+2}\) is
(A)\(-(x^{2}-2x+2)^{-2}\)
(B)\(ln(x^{2}-2x+2)\)
(C)\(ln\left | \frac{x-2}{x+1} \right |\)
(D)\( arcsec(x- 1)\)
(E)\( arctan(x- 1) \)
▶️Answer/Explanation
Ans:E
\(\int \frac{1}{x^2-2x+2}dx=\int \frac{1}{(x^2-2x+1)+1}dx=\int \frac{1}{(x-1)^2+1}dx=tan^{-1}(x-1)+C\)
Question
If the second derivative of f is given by f “(x) 2x-cosx , which of the following could be f ( x) ?
(A)\(\frac{x^{3}}{3} +cosx+x+1\)
(B)\(\frac{x^{3}}{3} – cosx-x+1\)
(C)\(x^{3}+cosx-x+1\)
(D)\(x^{2}-sinx+1\)
(E)\(x^{2}+sinx+1\)
▶️Answer/Explanation
Ans:A
\(f'(x)=x^2-sinx+C,f(x)=\frac{1}{3}x^3+cosx+Cx+K\) Option A is the only one with this form.