so the problem is
∫(sin7x)dx
you have to use u-substitution,
set u=7x
then du/dx=7 so du=7dx and solve for dx 1/7du=dx,
so you get
1/7∫(sinu)du then integrate like normal, and you get
1/7(-cosu)+C then you plug u back in and get 1/7(-cos7x)+C
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integrate(x5x dx) simplifies to integrate(5x^2 dx), and using the power rule of integration, add one to the power of x and divide the term by that number. Thus, x5x dx integrated is (5/3)x^3
Simply integrate all the pieces apart, en add them up. This is allowed, because int_a^c f(x)dx = int_a^b f(x)dx + int_b^c f(x)dx for all a,b,c in dom(f).
x-1 = 1/x ∫1/x dx = ln x + C
First, antiderivative = a solution to the indefinite integral therefore to integrate -(csc(x))(cot(x)) first convert it to -cos(x)/sin2(x) To integrate ∫-cos(x)/sin2(x) dx, use substitution u = sin(x) and du/dx = cosx This will make it ∫-1/u2 du and the antiderivative is 1/u +c, therefore the answer is 1/sin(x) + c.
The answer depends on what information you start with. For example, if you are given acceleration then you might integrate whereas if you are given displacement, you might differentiate.