Integration - Algebraic Substitution, 3

Category: Integral Calculus, Algebra

"Published in Suisun City, California, USA"

Evaluate

Solution:

Consider the given equation above

In this type of equation, we cannot integrate it by a simple integration because both the numerator and the denominator have radical equations. If you will rationalize the denominator in order to eliminate the radical sign, then the numerator still has radical equations and the above equation will be more complicated. To eliminate the radical signs at the numerator and the denominator, we have to use the Algebraic Substitution as follows

If 

then

 

 
Substitute the values of √x , √x + 1, and dx to the above equation, we have

but

Hence, the above equation becomes

Therefore, the final answer is

Derivative - Exponential Functions, 2

Category: Differential Calculus, Algebra

"Published in Newark, California, USA"

Find the derivative for

Solution:

Consider the given equation above

The above equation is an exponential function but we cannot take the derivative by exponential function or even by power because the exponent of an algebraic function is also an algebraic function. In this type of exponential function, we need to take natural logarithm on both sides of an equation in order to eliminate the algebraic exponent of an algebraic function. 

Take natural logarithm on both sides on an equation, we have

Take the derivative on both sides of the equation with respect to x

Multiply on both sides of the equation by y

but 

Hence, the above equation becomes

Therefore, the final answer is

Integration - Exponential Functions

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation above

Since 3 and e have the same exponent which is x, then we can rewrite the above equation as follows

If

then

Apply the Integration of Exponential Functions to the above equation, we have

Apply the Laws of Logarithm for the product of coefficients at the denominator

Therefore, the final answer is