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 \(\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x\) is

(a)

cot(xe^x)+c

(b)

sec(xe^x)+c

(c)

tan(xe^x)+c

(d)

cos(xe^x)+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

\(\int \frac{e^x\left(x^2 \tan ^{-1} x+\tan ^{-1} x+1\right)}{x^2+1} d x\) is

(a)

e^x tan^-1(x+1)+c

(b)

tan^-1(e^x)+c

(c)

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

(d)

e^x tan^-1 x+c

\(\int \sqrt{\frac{1-x}{1+x}} d x\) is

(a)

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

(b)

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

(c)

\(log|x+\sqrt{1-x^2}|-\sqrt{1-x^2}+c\)

(d)

\(\sqrt{1-x^2}+log|x+\sqrt{1-x^2}|+c\)

Integrate the following with respect to x : \({sin \ x \over cos ^2 \ x}\)

Integrate the following with respect to x : x¹¹

Integrate the following with respect to x : \({1\over \sqrt{1-25 x^2}}\)

Integrate the following functions with respect to x : cos (5 - 11x)

If f'(x) = 3x² - 4x + 5 and f(1) = 3, then find f(x).

Evaluate : \(\int {3x+7\over x^2-3x+2}dx\)

Integrate the following functions with respect to x : \({1\over \sqrt{1-81x^2}}\)

Integrate the following functions with respect to x : \(\frac{8^{1+x}+4^{1-x}}{2^x}\)