Integration - Trigonometric Functions, 9

Category: Integral Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation above

The given equation above has both trigonometric functions at the numerator and the denominator. The denominator contains logarithmic function of a trigonometric function. We need to assign u by considering the complicated part in the equation.

If 

then it follows that

 

Since the numerator has the same du, then the above equation can be integrated by simple integration as follows

Integration - Exponential Functions, 2

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation above

Since the numerator is greater than the denominator, then we have to do the division of polynomial as follows

The above equation becomes

If

then

Therefore,

 

Integration - Algebraic Substitution, 4

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate:

Solution:

Consider the above equation

The above equation cannot be integrated by simple integration because one of the variables has a radical sign. We need to eliminate the radical sign by substituting with another variable as follows

Let

So that the above equation becomes

but 

Therefore,

 

Integration - Partial Fractions

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation above 

if

then

Since the numerator is only dx, then we cannot integrate the above equation by simple integration. However, we can factor the denominator as follows

The above equation can be expressed into partial fractions as follows 

Consider

Multiply both sides of the equation by their Least Common Denominator (LCD), we have
 

 

 

  Equate x²:

Equate x:

Equate x0:

Substitute the values of A, B, and C to the original equation, we have

You can also simplify further the above equation as follows