Why is learning physics useful?
Besides the applications used by those who enter career fields that directly use physics to do research, solve problems, or engineer something, I think it is useful to learn some degree of physics. One may never need to directly know how long it takes a ball dropped from a building to hit the ground, what force two masses will have on each other, or how much voltage will pass through a particular resistor in a circuit, but I think physics teachers more than the equations to solve problems like those. I think that learning physics teaches one basic yet practical problem solving skills that apply generally across any sort of scientific or mathematical field. The practice of solving such problems does more than teach how to find answers to certain questions. Physics often times provides problems with no obviously apparent way to solve them, requiring one to use different approaches than he or she is accustomed to. Learning to look at problems in different ways and learning different approaches to solving problems is a useful tool in any analytic field, and is a skill that I think is learned very effectively through the study of physics.
Why should one participate in music?
As I progressed through high school, I noticed that an increasing number of students were dropping music classes (band, orchestra, choir, etc.) in order to take a wider range of academic classes. While I wholeheartedly agree that it is important to make academics a priority, I found myself confused as to why increasing numbers of people were musical pursuits from their activities. Despite having many difficult classes throughout high school, I deemed my musical pursuits as essential to my ability to function at a high level. While arguments could be made for any hobby or extracurricular activity as being necessary for a high level of functioning in school, I believe the arguments that can be made for music have extra merit. Becoming a proficient musician requires a large amount of time and effort spent practicing. A musician then learns the skill of dedication and hard work. At the onset, learning music and an instrument simultaneously provides a very steep learning curve. A musician then learns perseverance through challenges and adversity. Mastery of music at any level requires incredible attention to detail. A musician then learns concentration and to face a given task with maximum effort. While there are many more skills one can learn acquire from learning music, I believe that music is more than just a hobby or a means to acquire a skill. I believe that music is the purest way that emotions can be expressed, whether through performance or composition. It is because of this, I think, that listening to music is such a large part of so many people's lives, and also why I think that everyone should have the chance to participate in its production.
Compute the following indefinite integral ∫arctanx/x^2 dx
∫arctanx/x^2 dx Use integration by parts ∫udv = uv - ∫vdu where u = arctanx; dv = 1/x^2; du = 1/(x^2 + 1); v = -1/x The integral is then equivalent to: -arctanx/x + ∫1/(x(x^2 + 1))dx Note that 1 = 1 + x^2 - x^2; The fraction in the integral above then can be factored to: 1/x - x^2/(x(x^2 + 1)) which can be simplified to 1/x - x/(x^2 + 1). This leaves us with -arctanx/x + ∫1/x dx - ∫x/(x^2 + 1) dx; The computation of the first integral is lnx + C; For the second integral we use the u substitution u = x^2 + 1; then du = 2xdx. The second integral then reduces to -∫1 /(2u) du; evaluated, this gives us - 1/2 * lnu; substituting u = x^2 + 1 returns 1/2 * ln(x^2 + 1) + C. Putting all the pieces together, the solution then is -arctanx/x + lnx - 1/2 * ln(x^2 + 1) + C.