Pembahasan Soal UTS Kuliah Kalkulus Integral

Hello Guys

Welcome to my blog.
In this section we will try to solve mid term examination even semester 2019/2020 in my college, so let's look at the disscussion.

Question :

1. What is an indefinite integral? How do you evaluate one? What general formulas do you know for finding indefinite integrals? You may elaborate your answer with example(s)
Answer : Indefinite integral is an antiderivative of a function. Indefinite integral of function $f(x)$ is a function $F(x)$, such that $F'(x)=f(x)$. We can evaluate that by we find a function, $F(x)$, whose the derivative of this is $f(x)$.

General formula : 

$\int f(x)\ \text{dx}=F(x)+C$ , where $C$ is a constant and $F'(x)=f(x)$
2.  If you know the acceleration of a body moving along a coordinate line as a function of time, what more do you need to know to find the function of body’s position? Give an example

Answer : We need the value of the velocity of a body moving at a certain time or when the time is equal zero. And we also know that the position of a body at a certain time or when the time is equal zero.

Example : Given the function of acceleration of a body moving is satisfy

$a(t)=t^2+3t-1$

where at $t=0$, the velocity is $2$ and the position is $0$
3. Graph each function given over the given interval. Then, Partition the interval into subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum

           $\sum_1^n f(c_k).\Delta x_k, n > 4$
  a. $f(x)=1-x$,   $[0,2]$

Answer :

  b. $g(x)=sin\ x+\frac{1}{2}$,    $[-\pi,\pi]$

Answer :

  c. What is the relation between definite integrals and area? Describe your interpretation of definite integrals by using example.

Answer : Definite integrals is represent area of function between two curves and belong $[a,b]$ when the function is positive. And when the function is negatif definite integral is negative from the area. For example :

$\int_{-1}^1 x^3\ \text{dx}$

if we plot the graph $x^3$ is positive when $x>0$ and negative when $x<0$.

So,

$\int_{-1}^1 x^3\ \text{dx}$ is the area of $f(x)$ and $x$-axis between $[0,1]$ minus the area of $f(x)$ and $x$-axis between $[-1,0]$.
4. Is the following mathematical equation correct? If Yes, prove it and give an example of how to use this formula. If No, disprove it by giving counterexample(s) to show that it is incorrect.

            $\int_{a}^b f(x) dx=\int_{b}^a f((a+b)-x) dx$

Answer : No, this equation is incorrect. For example :

Let : $f(x)=2x$, $a=1$ and $b=2$ 

we have $F(x)=x^2+C$

Hence,

$\int_{a}^b f(x) dx=\int_{1}^2 2x dx$

$\int_{a}^b f(x) dx=3$

and then

$f((a+b)-x)=2(3-x)=6-2x$

Hence,

$\int_{b}^a f((a+b)-x) dx=\int_{2}^1 6-2x dx$

$\int_{b}^a f((a+b)-x) dx=-3$

Therefore

$\int_{a}^b f(x) dx\neq\int_{b}^a f((a+b)-x) dx$

5. Evaluate the following integral
   a.       $\int \frac{ay^2+by+c}{y^2(y^2+1)^2} dy$
   with $a,b,c$ respectively indicates the last digit number of your student number (NIM), the last digit your number of birth date, and any real number you choose.

Answer : set : $a=2,b=8,c=0$

we have the following integral

$\int \frac{2y^2+8y}{y^2(y^2+1)^2} dy$

$=\int \frac{2y+8}{y(y^2+1)^2} dy$

Note that :

$\frac{2y+8}{y(y^2+1)^2}=\frac{8}{y}-\frac{8y}{y^2+1}+\frac{-8y+2}{(y^2+1)^2}$

So, we get

$\int \frac{2y+8}{y(y^2+1)^2} dy$

$=\int \frac{8}{y} dy-\int \frac{8y}{y^2+1} dy+\int \frac{-8y+2}{(y^2+1)^2} dy$

$=4ln\left(\frac{y^2}{y^2+1}\right)-\frac{y+4}{y^2+1}-tan^{-1}y+C$

   c.        $\int cos\ \sqrt{ax+b} dx$
   with $a$ and $b$ respectively indicates different real number you choose, in which $a\neq 0$, $b\neq 0$

Answer : Set $a=1$ and $b=-1$

Now let : $x-1=y^2$

                $dx=2y\ dy$

so, we have the following integral

$\int cos\ y .2y\ dy$

$=2\int y.cos\ y \ dy$

With integration by parts, we get

$=2\left(y.sin\ y+cos\ y+C\right)$

$=\sqrt{4x-4}sin \sqrt{x-1}+2cos \sqrt{x-1}+C$

Oke guys, maybe that's enough for today, and see you next day. If you have any questions please write in the comments. Oke thank you guys.