Evaluate the indefinite integral of: ∫ (x^4 + 4x^2 +5)(x^3 + 2x) dx
We will have to do u-substitution for this problem, with u =x^4 + 4x^2 +5 Then, take the derivative of u: du = 4x^3 +8x dx Next, factor du: du = 4(x^3 + 2) dx Lastly, solve for dx = du / 4(x^3 + 2) Then, substitute dx back into the equation, and extract the constant: (x^3 + 2x)dx = 1/4 * du = 1/4 ∫ u * du Then, use the power rule to evaluate the indefinite integral: 1/4 * (1/2)u^2 + C Last step, re-substitute your u: (1/8)(x^4 + 4x^2 + 5)^2 + C
If sin(xy) = x, then dy/dx = ?
First, we must observe that dy/dx stands for the derivative of y, with respect to x. Next, we have to notice that there are two variables in the expression (x and y). Since there are two variables, we will have to use implicit differentiation, with respect to x. Also, because sin(u) is the outside function, with u=xy as the inner function, we need to use the chain rule. As a reminder, if you have an outside function that contains an inside function (like in our case), you can differentiate with the chain rule in the following manner: 1. Take the derivative of the outer function, and keep the inner function inside the parenthesis 2. Take the derivative of the inner function, and multiply it by what you have in step 1 ***In our case**** 1. The derivative of the outer function Sine, is equal to Cosine, and our inside function is (xy), so we keep the inner function inside the derivative ------------> cos(xy) 2. The derivative of the inner function, xy, will be calculated using implicit differentiation. In our case, we want the derivative with respect to x --------------> y + x(dy/dx) Putting this together gives us: cos(xy) * [y + x(dy/dx)] Going back to the original equation we were given, we also have to differentiate the right side of the equation as well - since we already differentiated the left side. Since the derivative of x is just 1, we can now write the equation like: cos(xy) * [y + x(dy/dx)] = 1 Now, all we have to do is solve for dy/dx Fortunately, this is very easy algebra at this point dy/dx = (1 - y*cos(xy)) / (x*cos(xy))
What are the three most commonly used types of supports in a truss? Describe the full body diagram reactions for each type.
1. Fixed end support: This support is braced to prevent horizontal & vertical translation, as well as clockwise and counterclockwise rotation. There are three reactions for the full body diagram: a force in the horizontal direction, a force in the vertical direction, and a moment. 2. Pinned support: This support is braced to prevent horizontal & vertical translation; however, it is not braced to prevent clockwise or counterclockwise rotation. There are two reactions for the full body diagram: a force in the horizontal direction, and a force in the vertical direction. 3. Roller support: This support is braced to only prevent vertical translation. Therefore, it only has one reaction for the full body diagram: a force in the vertical direction.