Variable Separation, 2

Category: Differential Equations, Integral Calculus

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above

Divide both sides of the equation by tan x tan y, we have

Integrate both sides of the equation

Apply the laws of logarithm for the above equation, we have

Take the inverse natural logarithm on both sides of the equation

Therefore,

Solving Logarithmic Equation, 2

Category: Algebra

"Published in Suisun City, California, USA"

Solve for the value of x for

Solution:

Consider the given equation above

Apply the Laws of Logarithm for the given equation above, we have

Take the inverse logarithm on both sides of the equation to the base 2, we have

Check:

Improper Integral

Category: Integral Calculus

"Published in Suisun City, California, USA"

Evaluate

Solution:

The given equation above is Improper Integral where the limits are - ∞ and + ∞. In this case, we can split the limits as follows

Consider

Since the lower limit is -∞, we can rewrite the above equation as follows

Consider

Since the upper limit is +∞, we can rewrite the above equation as follows

Therefore,

Since the given Improper Integral has a value, then the given Improper Integral is convergent.