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#### 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...

#### 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...

#### Integration By Parts

In this example, there is a product of x^2 and cosx dx in the integrand. Does simple u-substitution method work for this example? In order to use the simple u-substitution method, the relationship between two functions must be the original and its derivative each other except for constants. Unfortunately, x^2 cannot be the derivative of...