Integrals & Trig Functions

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How to Use Tangent Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of the following forms: [more…]

How to Use Sine Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of certain forms because they match up with trigonometric functions. A sine can take the place of a radical in a particular [more…]

How to Integrate Sine/Cosine Problems with an Odd, Positive Power of Cosine

When you integrate a trig integral that includes cosine, and if the power of cosine is odd and positive, you can convert and then use substitution to integrate. To make this conversion, you need to know [more…]

How to Integrate Tangent/Secant Problems with an Odd, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents and secants where the tangent power is odd and positive. You’ll need the tangent-secant version of the Pythagorean identity: [more…]

How to Integrate Tangent/Secant Problems with an Even, Positive Power of Secant

Here’s how you integrate a trig integral that contains tangents and secants where the secant power is even and positive. Like with all tangent/secant integrals, you use the tangent-secant version of the [more…]

How to Integrate Problems with an Even, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents (and no secant factors) where the tangent power is even and positive. [more…]

How to Use Trig Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of the following forms (given a is a constant and u is an expression containing [more…]

How to Integrate Sine/Cosine Problems with an Odd, Positive Power of Sine

Here’s how you integrate a trig integral that contains sines and cosines where the power of sine is odd and positive. You lop off one sine factor and put it to the right of the rest of the expression, [more…]

How to Integrate Sine/Cosine Problems with Even, Nonnegative Powers of Both Sine and Cosine

Here’s how you integrate a trig integral that contains sines and cosines where the powers of both sine and cosine are even and nonnegative (in other words, zero or positive). You first convert the integrand [more…]

How to Solve Integrals with Variable Substitution

In Calculus, you can use variable substitution to evaluate a complex integral. Variable substitution allows you to integrate when the Sum Rule, Constant Multiple Rule, and Power Rule don’t work. [more…]

How to Use Trig Substitution to Integrate radicals of the sine form

Before reading this article, you should check out the discussion of trig substitution in the companion article, “How to Use Trig Substitution to Integrate.” [more…]

How Circuit Analysis Works in the s-Domain

Circuit analysis techniques in the s-domain are powerful because you can treat a circuit that has voltage and current signals changing with time as though it were a resistor-only circuit. That means you [more…]

Making Low-Pass and High-Pass Filters with RC Circuits

With simple RC circuits, you can build first-order RC low-pass (LPF) and high-pass filters (HPF). These simple circuits can give you a foundational understanding of how filters work so you can build more-complex [more…]

Create Band-Pass and Band-Reject Filters with RLC Series Circuits

There are many applications for an RLC circuit, including band-pass filters, band-reject filters, and low-/high-pass filters. You can use series and parallel RLC circuits to create band-pass and band-reject [more…]

Create Band-Pass and Band-Reject Filters with RLC Parallel Circuits

There are many applications for an RLC circuit, including band-pass filters, band-reject filters, and low-/high-pass filters. You can use series and parallel RLC circuits to create band-pass and band-reject [more…]

Laplace Transforms and s-Domain Circuit Analysis

Laplace transform methods can be employed to study circuits in the s-domain. Laplace techniques convert circuits with voltage and current signals that change with time to the [more…]

Describe Second-Order Circuits with Second-Order Differential Equations

If you can use a second-order differential equation to describe the circuit you’re looking at, then you’re dealing with a second-order circuit. Circuits that include an inductor, capacitor, and resistor [more…]

s-Domain Analysis: Understanding Poles and Zeros of F(s)

Laplace transforms can be used to predict a circuit's behavior. The Laplace transform takes a time-domain function f(t), and transforms it into the function [more…]

Analyze a First-Order RC Circuit Using Laplace Methods

Using the Laplace transform as part of your circuit analysis provides you with a prediction of circuit response. Analyze the poles of the Laplace transform to get a general idea of output behavior. Real [more…]

Analyze an RLC Circuit Using Laplace Methods

Using the Laplace transform as part of your circuit analysis provides you with a prediction of circuit response. Analyze the poles of the Laplace transform to get a general idea of output behavior. Real [more…]

Analyze a First-Order RL Circuit Using Laplace Methods

Using the Laplace transform as part of your circuit analysis provides you with a prediction of circuit response. Analyze the poles of the Laplace transform to get a general idea of output behavior. Real [more…]

Generalize Impedance to Expand Ohm’s Law to Capacitors and Inductors

Use the concept of impedance to gernalize Ohm’s law in phasor form so you can apply and extend the law to capacitors and inductors. After describing impedance, you use phasor diagrams to show the phase [more…]

How to Use Phasors for Circuit Analysis

A phasor is a complex number in polar form that you can apply to circuit analysis. When you plot the amplitude and phase shift of a sinusoid in a complex plane, you form a phase vector, or phasor. [more…]

Analyze an RLC Second-Order Parallel Circuit Using Duality

Second-order RLC circuits have a resistor, inductor, and capacitor connected serially or in parallel. To analyze a second-order parallel circuit, you follow the same process for analyzing an RLC series [more…]

Analyze a Series RC Circuit Using a Differential Equation

A first-order RC series circuit has one resistor (or network of resistors) and one capacitor connected in series. First-order RC circuits can be analyzed using first-order differential equations. By analyzing [more…]

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