DivisionByZro said:Show us what you've got. Here are hints:
1 - Use the identity that cos(x)=2*cos(x/2)-1, and then use a simple substitution.
2 - Somewhat similar to the first one.
That identity should beDivisionByZro said:Show us what you've got. Here are hints:
1 - Use the identity that cos(x)=2*cos(x/2)-1, and then use a simple substitution.
2 - Somewhat similar to the first one.
The ∫(x*sec^2(x)).dx + 2∫tan(x/2).dx that you have should be: ∫(x*sec^2(x/2)).dx + 2∫tan(x/2).dx .Redoctober said:I got tan(x/2) + C for the first But nothin for the second .
i did as follows
∫(x+sin(x))/(x+cos(x)).dx
simplified it to
∫(x*sec^2(x)).dx + 2∫tan(x/2).dx
I then i integrated them , i used integration by parts for then right hand integral
Then i finally got
2x*tan(x/2)
Is it correct :D ?
SammyS said:The ∫(x*sec^2(x)).dx + 2∫tan(x/2).dx that you have should be: ∫(x*sec^2(x/2)).dx + 2∫tan(x/2).dx .
Yes, your answer is correct, if you add an arbitrary constant.
Integration is a mathematical process of finding the total or sum of a quantity by adding up infinitely small pieces. It is important because it allows us to solve problems involving continuous change, such as finding areas, volumes, and displacement.
The most commonly used methods of integration are the following: integration by substitution, integration by parts, partial fractions, and trigonometric substitution. These methods allow us to solve a variety of integration problems by using different techniques.
The choice of method depends on the form of the integrand (the function being integrated). Generally, we try to manipulate the integrand to make it easier to integrate using one of the methods mentioned above. This requires practice and familiarity with the different techniques.
Yes, integration can be used to find the area under a curve. This is known as the definite integral, where we specify the limits of integration (the start and end points) to find the exact area between the curve and the x-axis.
You can check your answer by taking the derivative of the integrated function. If the derivative matches the original function, then your answer is correct. You can also use online integration calculators to verify your answer or solve the problem step-by-step.