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QUESTION 1 1.1 Integrate the following in terms of x: $\int \left(\frac{1}{\sqrt{2x}} + 2^{-10x} + \frac{e^{-kx}}{2} - 4x^2 + 2\pi \cos wx - 5 \cos ec^2 5x\right) dx$ 1.2 Evaluate: $\int_0^{\frac{\pi}{4}} \sin 2x dx$ 1.3 1.3.1 Draw the graph of $y = 2^x$, the X-axis, $x = 1$ and $x = 4$ and clearly indicate the enclosed area. Also show the representative strip that will be used to calculate the enclosed area. 1.3.2 Determine, using integration, the value of the enclosed area. 1.4 Simplify: $\int (4e^x - 11x^3) dx$ QUESTION 2 2.1 Expand $\frac{1}{\sqrt{16x^2 + 4}}$ to four terms only by the use of the binomial theorem. 2.2 Differentiate the following in terms of x: $y = \frac{5 + 6\sin x + x \cos x}{- \cos x} + e^{-6} - 2 \log_e x$ 2.3 Determine, with the aid of differentiation, the coordinates of the maximum and minimum turning points of $y = x^2 - 2x - 4$. Also distinguish between the maximum and the minimum turning points by use of the second derivative. 2.4 Differentiate the following by using the quotient rule: $y = \frac{\ln x^3}{\ln x}$

          QUESTION 1
1.1 Integrate the following in terms of x:
$\int \left(\frac{1}{\sqrt{2x}} + 2^{-10x} + \frac{e^{-kx}}{2} - 4x^2 + 2\pi \cos wx - 5 \cos ec^2 5x\right) dx$
1.2 Evaluate:
$\int_0^{\frac{\pi}{4}} \sin 2x dx$
1.3 1.3.1 Draw the graph of $y = 2^x$, the X-axis, $x = 1$ and $x = 4$ and clearly indicate
the enclosed area. Also show the representative strip that will be used to
calculate the enclosed area.
1.3.2 Determine, using integration, the value of the enclosed area.
1.4 Simplify:
$\int (4e^x - 11x^3) dx$
QUESTION 2
2.1 Expand $\frac{1}{\sqrt{16x^2 + 4}}$ to four terms only by the use of the binomial theorem.
2.2 Differentiate the following in terms of x:
$y = \frac{5 + 6\sin x + x \cos x}{- \cos x} + e^{-6} - 2 \log_e x$
2.3 Determine, with the aid of differentiation, the coordinates of the maximum and
minimum turning points of $y = x^2 - 2x - 4$. Also distinguish between the maximum
and the minimum turning points by use of the second derivative.
2.4 Differentiate the following by using the quotient rule:
$y = \frac{\ln x^3}{\ln x}$

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QUESTION 1
1.1 Integrate the following in terms of x:
∫((1)/(√(2x)) + 2^-10x + (e^-kx)/(2) - 4x^2 + 2πcos wx - 5 cos ec^2 5x) dx
1.2 Evaluate:
∫0^(π)/(4)sin 2x dx
1.3 1.3.1 Draw the graph of y = 2^x, the X-axis, x = 1 and x = 4 and clearly indicate
the enclosed area. Also show the representative strip that will be used to
calculate the enclosed area.
1.3.2 Determine, using integration, the value of the enclosed area.
1.4 Simplify:
∫ (4e^x - 11x^3) dx
QUESTION 2
2.1 Expand (1)/(√(16x^2 + 4)) to four terms only by the use of the binomial theorem.
2.2 Differentiate the following in terms of x:
y = (5 + 6sin x + x cos x)/(- cos x) + e^-6 - 2 x
2.3 Determine, with the aid of differentiation, the coordinates of the maximum and
minimum turning points of y = x^2 - 2x - 4. Also distinguish between the maximum
and the minimum turning points by use of the second derivative.
2.4 Differentiate the following by using the quotient rule:
y = (ln x^3)/(ln x)

Added by Daniel W.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Vincenzo Zaccaro

08/07/2022

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expand Divide[1,Sqrt[Square[16t]]+4] to four terms only by the use of the binomial theorem (16030164) -3- T1020(E)(N16)T QUESTION1 1.1 Integrate the following in terms of x : -4+2cos wx-5cos ec5x 2 (7) 1.2 Evaluate: [sin2xdx (4) 1.3 1.3.1 Draw the graph of y=2,the X-axis, x=I and x=4 and clearly indicate the enclosed area. Also show the representative strip that will be used tc calculate the enclosed area. (2) 1.3.2 Determine, using integration, the value of the enclosed area (4) 1.4 Simplify: (3) [20] QUESTION2 2.1 1 Expand to four terms only by the use of the binomial theorem. 16t+4 (4) 2.2 Differentiate the following in terms of x: 5+6sin x+xcos x+e -2log, x -cosx (5) 2.3 Determine, with the aid of differentiation, the coordinates of the maximum and minimum turning points of y = 3 -2x -4. Also distinguish between the maximum and the minimum turning points by use of the second derivative. (7) 2.4 Differentiate the following by using the quotient rule: In x3 Inx (4) [20]

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00:01 Okay, so let's get started with the first one.

00:04 Well, the first one is the area under this curve and above the interval 0b.

00:13 And this area is just an integral from 0 to b of x over 4 plus 2 in the x.

00:25 And well, so clearly the right answer is a.

00:29 B squared over 8 plus 2b now let's find the derivative of this function well thanks to the fundamental theorem of calculus this one is 12 x to the 9 halfs multiplied by oh actually let me write this 9 okay so 9 halves multiplied by 1 over 2 x to the 1 half so the 1 half so the answer is this one here see now let's find the area of this shaded region well this area is what this area is an integral with respect to y between negative 3 and 3 of okay 9 minus y squared in the y and the y and this this guy here is okay let's compute it this one is gonna be nine y minus y cubed over three evaluated between negative three and three okay so let me perform this computation here so nine y minus y cubed over 3 evaluated between negative 3 and 3.

02:14 Well this one is the same thing as 2 multiplied by 9 y minus y cubed over 3 evaluated between 0 and 3.

02:29 So here we get 2 multiplied by 27 minus 27 over 3 so minus 9 and this guy here is 2 multiplied by 18 which is 36 so the right answer here is d now let's evaluate this integral so this one is going to be long so i'm gonna write here we're gonna we need to evaluate the integral of the x over x log of x to the 6th power so let's keep in mind that log of x to the sixth power is 6 log of x so here we are going to have 1 over 6 multiplied by an integral of 1 over x which is the derivative of log of x multiplied by 1 over x multiplied by 1 over log of x in the x now since this one is the derivative of log of x here we are going to have 1 over 6 log of log of x okay perfect so here okay i think here there is a typo that is this one should be log of x so it should be just log of x not log of x not log of x to the 6 power but anyway this is the right answer.

04:24 Okay, now let's do the last one.

04:28 Well, here we just need to notice that the derivative of x to the sixth power plus eight is six multiplied by x to the fifth power.

04:44 So, okay, now we can rewrite this one.

04:50 I'm gonna write here.

04:53 So x to the sixth power plus 8...

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