If \(\int f(x) d x=g(x)+c\), then \(\int f(x) g^{\prime}(x) d x\)

(a)

\(\int (f(x))^2dx\)

(b)

\(\int f(x)g(x)dx\)

(c)

\(\int f'(x)g(x)dx\)

(d)

\(\int (g(x))^2dx\)

If \(\int {3^{1\over x}\over x^2}dx=k(3^{1\over x})+c\), then the value of k is

(a)

log 3

(b)

-log 3

(c)

\(-{1\over log3}\)

(d)

\({1\over log3}\)

If \(\int f '(x)e^{x^2}dx=(x-1)e^{x^2}+c\), then f(x) is

(a)

\(2x^3-{x^2\over 2}+x+c\)

(b)

\({x^3\over 2}+3x^2+4x+c\)

(c)

x³ + 4x² + 6x + c

(d)

\({2x^3\over3}-x^2+x+c\)

The gradient (slope) of a curve at any point (x, y) is\({x^2-4\over x^2}\). If the curve passes through the point (2, 7), then the equation of the curve is

(a)

\(y=x+{4\over x}+3\)

(b)

\(y=x+{4\over x}+4\)

(c)

y = x² + 3x + 4

(d)

y = x² - 3x + 6

\(\int \sin ^3 x d x\) is

(a)

\({-3\over 4}cos \ x-{cos \ 3x\over 12}+c\)

(b)

\({3\over 4}cos \ x+{cos \ 3x\over 12}+c\)

(c)

\({-3\over 4}cos \ x+{cos \ 3x\over 12}+c\)

(d)

\({-3\over 4}sin \ x-{sin \ 3x\over 12}+c\)

\(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{4 \log x}-e^{3 \log x}} d x\) is

(a)

x+c

(b)

\({x^3\over 3}+c\)

(c)

\({3\over x^3}+c\)

(d)

\({1\over x^2}+c\)

\(\int \frac{\sec x}{\sqrt{\cos 2 x}} d x\) is

(a)

tan^-1 (sin x)+c

(b)

2sin^-1(tan x)+c

(c)

tan^-1(cos x)+c

(d)

sin ^-1(tan x)+c

\(\int \tan ^{-1} \sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}} d x\) is

(a)

x²+c

(b)

2x²+c

(c)

\({x^2\over2}+c\)

(d)

\(-{x^2\over2}+c\)

\(\int 2^{3 x+5} d x\) is

(a)

\({3(2^{3x+5})\over log 2}+c\)

(b)

\({2^{3x+5}\over 2log (3x+5)}+c\)

(c)

\({2^{3x+5}\over 2log 3}+c\)

(d)

\({2^{3x+5}\over 3log 2}+c\)

\(\int \frac{\sin ^8 x-\cos ^8 x}{1-2 \sin ^2 x \cos ^2 x} d x\) is

(a)

\({1\over2}sin 2x+c\)

(b)

\(-{1\over2}sin 2x+c\)

(c)

\({1\over2}cos 2x+c\)

(d)

\(-{1\over2}cos 2x+c\)