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)

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

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

(a)

x+c

(b)

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

(c)

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

(d)

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

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

(a)

x²+c

(b)

2x²+c

(c)

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

(d)

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

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

(a)

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

(b)

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

(c)

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

(d)

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

\(\int \frac{x^2+\cos ^2 x}{x^2+1} \operatorname{cosec}^2 x d x\) is

(a)

cot x + sin ^-1x + c

(b)

-cot x + tan^-1x + c

(c)

-tan x + cot^-1x + c

(d)

-cot x - tan^-1x + 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)

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

\(\int \frac{d x}{e^x-1}\) is

(a)

\(log|e^x|-log|e^x-1|+c\)

(b)

\(log|e^x|+log|e^x-1|+c\)

(c)

\(log|e^x-1|-log|e^x|+c\)

(d)

\(log|e^x+1|-log|e^x|+c\)

\(\int x^2 e^{\frac{x}{2}} d x\) is

(a)

\( x^2e^{x\over2}-4xe^{x\over2}-8e^{x\over2}+c\)

(b)

\( 2x^2e^{x\over2}-8xe^{x\over2}-16e^{x\over2}+c\)

(c)

\( 2x^2e^{x\over2}-8xe^{x\over2}+16e^{x\over2}+c\)

(d)

\( x^2{e^{x\over2}\over 2}-{xe^{x\over2}\over 4}+{e^{x\over2}\over 8}+c\)

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

(a)

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

(b)

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

(c)

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

(d)

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

\(\int e^{\sqrt{x}} d x\) is

(a)

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

(b)

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

(c)

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

(d)

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

Integrate the following with respect to x : x¹⁰

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

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

Integrate the following with respect to x : \({1\over (3x-2)}\)

If f '(x) = 4x - 5 and f(2) = 1, find f(x).

Evaluate : \(\int e^{xlog2}e^x dx\)

Integrate the following functions with respect to x : (2x − 5)(36 + 4x)

Integrate the following with respect to x : e^ax cos bx

Evaluate the following integrals : \(\int{1\over (x+2)^2+1}dx\)

Integrate the following with respect to x : 12³

Integrate the following functions with respect to x : e^{3x - 6}

Evaluate the following integrals : \({15\over \sqrt{5x-4}}-8cot (4x+2)cosec(4x+2)\)

Evaluate :\(\int(tan \ x+cot \ x)^2dx\)

Integrate the following functions with respect to x : \({1+cos \ 4x \over cot \ x-tan \ x}\)

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

A tree is growing so that, after t - years its height is increasing at a rate of \({18\over \sqrt{t}}\) cm per year Assume that when t = 0, the height is 5 cm.
(i) Find the height of the tree after 4 years.
(ii) After how many years will the height be 149 cm?

Integrate the following functions with respect to x : \(x+1\over (x+2)(x+3)\)