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Prove that the indefinite integral of I = int(exp(x).cos(x))dx is (1/2)exp(x).sin(x) + (1/2)exp(x).cos(x) + C

Starting with the initial integral of int(exp(x).cos(x))dx we can see that this is going to have to be integrated by parts. This states that the integral of (u . dv/dx)dx is equal to u.v - int(v . du/dx)dx T...
SA
Answered by Sammy A. • Maths tutor
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A curve has the equation y = x^4 - 8x^2 + 60x + 7. What is the gradient of the curve when x = 6?

To find the gradient of any curve, we take the derivative. So in this case, we need to take dy/dx. We do this by multiplying the term by the power on x, and then lowering the power by one. For example, for t...
EH
Answered by Elizabeth H. • Maths tutor
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Integrate xsin(x) by parts between the limits of -pi/2 and +pi/2

Let u = x and dv/dx = sin(x), By using the general expression of: integral(u multiply dv/dx)dx = [u multiply v] - integral(v multiply du/dx)dx, and by realising that: du/dx = 1, and v = -cos(x), we can deduc...
MB
Answered by Matthew B. • Maths tutor
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Differentiate x^2+6x+1

All we do here is break down into three parts: x 2 , 6x & 1.x 2 becomes 2x as we multiply by the power and then decrease the power by one.6x becomes 6 and 1 becomes 0.So alltogether we have2x+6
SI
Answered by Samuel I. • Maths tutor
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find the definite integral between limits 1 and 2 of (4x^3+1)/(x^4+x) with respect to x

first notice the integral is in the form f'(x)/f(x), and indefinite integrals of this form are ln|f(x)|+c. therefore the integral is [ln|x 4 +x|] between limits 1 and 2. subbing in limits gives ln|2 4 +2|-ln...
TD
Answered by Tutor22645 D. • Maths tutor
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