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Sunday, June 2, 2013

More Integration Procedures, 9

Category: Integral Calculus, Trigonometry

"Published in Suisun City, California, USA"

Evaluate


Solution:

Consider the given equation above


There are two functions in the given equation which are x3 and sin x, respectively. If

 then

If
then

By applying the integration by parts, we have







Since the second term at the right side of the equation have two functions which are x2 and cos x, then we have to apply the integration by parts again. If

then

If
then

Again, by applying the integration by parts, we have







Therefore, the first integration by parts becomes







Since the second term at the right side of the equation have two functions which are x and sin x, then we have to apply the integration by parts again. If

then

If
then

Again, by applying the integration by parts, we have

 


 



Therefore, the final answer is









                                                                                                 

Saturday, June 1, 2013

Indeterminate Form - Zero Over Zero, 3

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Evaluate


Solution:

Consider the given equation above



Substitute the value of x which is 0 to the above equation, we have



Since the answer is 0/0, then it is considered as Indeterminate Form which is not accepted as a final answer in Mathematics. Remember this, any number, except zero divided by zero is always equal to infinity. If the Indeterminate form is either 0/0 or /, then we can use the L'Hopital's Rule in order to solve for these type of Indeterminate Forms. In this case, we can apply the L'Hopital's Rule for the above equation as follows



 

Substitute the value of x which is 0 to the above equation, we have


Since the answer is again 0/0, then we have to apply again the L'Hopital's Rule, as follows






Substitute the value of x which is 0 to the above equation, we have


Since the answer is again 0/0, then we have to apply again the L'Hopital's Rule, as follows




Substitute the value of x which is 0 to the above equation, we have


Therefore,