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Why does integration by parts work?

Let's consider the form of the formula we use for integration by parts: the integral of u * dv/dx = uv - the integral of v * du/dx. We know that integration is the inverse of differentiation, so we should be...
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Find the area enclosed by the curve y = cos(x) * e^x and the x-axis on the interval (-pi/2, pi/2)

To find the area enclosed by a function, we need to use integration. Here, the function is not easy to integrate just by inspection, but it is the product of two functions we do know how to integrate: cos(x)...
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Answered by Tomas S. • Maths tutor
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What is differentiation and how do I do it?

explain the following briefly: To work out the gradient, first principles, shortcutsexplain in detail: different methods
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Given that: y = 3x^2 + 6x^1/3 + (2x^3 - 7)/(3x^1/2), x > 0 Find dy/dx, give each term in its simplest form

y = 3x 2 + 6x 1/3 + (2x 3 /3 - 7/3)x -1/2 Using the multiplication rule for indices y = 3x 2 + 6x 1/3 + (2/3)x 5/2 - (7/3)x -1/2 Then differentiate each term with respect to x to find dy/dxdy/dx = 6x + 2x -2...
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a circle c has the equation x^2 + y^2 -4x + 10y = k. find the center of te circle

you would first change the equaton into the circle equation therefore completing the square (x -2)^2 -4 + (y + 5)^2 -25 = know use the equation to deduce the center of the circle (x -2)^2 + (y + 5)^2 = k + 2...
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