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#### Logarithmic Differentiation This method allows us to differentiate very complicated fractional functions or functions raised to the power of another function easily.

#### Integration by Simple u substitution #### Graphing by Hand

Here are tips for graphing. First, you need to find x-intercepts and y-intercept, if possible.  For x-intercepts, setting y=0 to find x-values and similarly, setting x=0 to find y-intercept. Then, investigate if there are any asymptotes for fractional functions. For the horizontal asymptotes, we need to take the limit at infinity.  If you get the...

#### Limits At Infinity To evaluate the limit as x approaches infinity, take the largest exponent term each  from the numerator and denominator, simplify the fraction, and then apply the limit rules as shown.

#### Integrating by Trigonometric Substitution This method allows us to change algebraic functions into trigonometric functions, integrate them in trigonometric forms, and return to the original algebraic functions as solutions.

#### Solving Differential Equation by Laplace Transforms Before starting the example, you need to know the following steps to solve DE by Laplace transforms. Step 1.  Take the Laplace transforms of both sides of the equation. Step 2. Solve for the Laplace of Y. Step 3. Manipulate the Laplace transform, F(s) until it matches one or more table entries. Step 4. Take...

#### Fourier Series Before looking at the example, you need to know the formula of Fourier Series as shown. If you know the additional information shown, you could reduce your work.  (1) Odd functions have Fourier Series with only sine terms, which means you only find the coefficients, b’s in the formula.  (2) Even functions have Fourier Series...

#### Derivatives of Logarithmic Functions Before starting examples, you need to know the derivative formulas as shown. In many cases, we need to make use of the properties of logarithm as well. Please remember that if you see “ln” symbol, it is called natural log and it has the base “e”. If you see “log”, it has the base 10.

#### 2nd order Nonhomogeneous Differential Equation Before looking at the example, you need to know the solution formula for second-order differential equations ay”+ by’ +cy = f(x) as shown.  Notice that there are two parts, y-sub C and y-sub P in the complete solution. One part, y-sub C is solving a homogeneous differential equation.  Y-sub C is often called a complementary...

#### Integrating Rational Fractions by Partial Fractions If you take a look at the integrand of the question, it seems a relatively complicated fraction.  If we can split it into simpler fractions, then we may be able to integrate them easily. Making use of partial fractions to get the simpler fractions. First, we need to have factored form of the denominator to...
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