Question

Differentiate the following functions with respect to x: (i) ln(1 + sin2 x) (ii) xx.
(c) Evaluate the integral 2x3 −4x−8 /x4 −x3 +4x2 −4x dx.

Answer 1

1.i. $ln [ 1 + sin 2x]$

Differentiating with respect to $x$ :

$d( ln [ 1 + sin2x])/dx$

$=[2 cos(2x)] / [ 1 + sin 2x]$ (Answer)

ii. $x^x$

Let A = $x^x$

Taking natural logarithm on both sides:

$ln A= x ln x$

Differentiating with respect to x:

$d[ ln A]/dx = d[x ln x] / dx$

or, $1 / A * d[A]/dx = x/x + ln x$

or, $dA/dx = (1 + ln x) * A$

Putting A = $x^x$

$dA/dx = [1 + ln x] * X^x Answer$

c. Solve $\int ( 2x^3 - 4x - 8/x^4 - x^3 + 4x^2 -4x) dx$

= $2x^4/4 - 4x^2/2 - 8x^{-3} /( -3) - x^4/4 + 4x^3/3 - 4x^2/2 + c$

where C is an integration constant

= $x^4/2 - 2x^2 + 8(x^-3 / 3) - x^4/4 + 4x^3/3 - 2x^2 + c$

=$[x^4/2 - x^4/4] - [ 2x^2 + 2 x^2] + 8x^-3/3 + 4x^3/3 + c$

=$[x^4/4] - 4x^2 + 8/3x^3 + 4x^3/3 + c [Answer]$