Integration Procedure - Algebraic Susbstitution

Category: Integral Calculus, Algebra, Differential Calculus

"Published in Newark, California, USA"

Evaluate

Solution:

If you examine the given equation, the denominator is complicated because it has a variable and a radical function as well. The differential at the numerator is not match with the differential of the denominator. Therefore, we cannot integrate the given equation by simple integration. If you will integrate the given equation by parts, then the equation will be more complicated and there's no end in integration as well. We have a procedure for that type of integration which is called an algebraic substitution. Let's consider the given equation

Let

Substitute these values to the given equation, we have

But

The equation becomes

Since the first term is indeterminate form, then we have to evaluate the limit of the first term since ∞/∞ is not accepted as a final answer in Mathematics. Let's evaluate the limit of the first term as follows

Apply the L'Hopital's Rule

Therefore,

Rationalize the denominator by multiplying both sides of the fraction by the conjugate of the denominator, we have

Solving Exponential Equation

Category: Algebra

"Published in Suisun City, California, USA

Find the value of x for

Solution:

Consider the give equation

Rewrite the above equation as follows

Take the natural logarithm on both sides of the equation

Therefore,

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Simplifying Algebraic Expression - Radical Group

Category: Algebra

"Published in Suisun City, California, USA"

Simplify the expression for:

Solution:

Consider the given equation

Square the term first inside the parenthesis

Get the LCD of the two terms inside the radical sign and combine all the terms in the numerator

Rationalize the denominator in order to remove the radical sign by multiplying both sides of the fraction by the denominator, we have