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How do you integrate sin(3x)cos(5x)?

STEP 1 Cannot integrate directly in this form, therefore use a trigonometric identity. Identity: sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2) STEP 2 (A+B)/2 = 3x A + B = 6x (1) (A-B)/2 = 5x A - B = 10x (2) ST...
CF
Answered by Catherine F. • Maths tutor
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Why does the constant disappear when differentiating a function?

We can think of the constant term in a function in terms of x, for example in x^2 + 3x + 2 as 2 being multiplied by x^0. Anything to the power of 0 is equal to one, so in our example we would have 2 * x^ 0 w...
AP
Answered by Abdullah P. • Maths tutor
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Can you show me why the integral of 1/x is the natural log of x?

We can break this down into steps, going deeper each time. First we might just say: well, since integration is fundamentally the inverse process of differentiation and we know that the derivative of ln(x) is...
JL
Answered by James L. • Maths tutor
10920 Views

(C3 question). Find an expression for all stationary points on the curve y=sin(x)cos(x). How many such points are there and why?

Stationary points are points at which the gradient of the curve is zero. The gradient is given by dy/dx so we start by computing this using product rule to give us dy/dx=-sin 2 x+cos 2 x. Notice that this is...
TG
Answered by Tristan G. • Maths tutor
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(FP3 question). Integrate 1/sqrt(3-4x-x^2).

There are two stages to this integral. Stage 1: Notice that there is a quadratic inside a square root in the denominator. We wish to look for a substitution which will reduce this problem to a known integral...
TG
Answered by Tristan G. • Maths tutor
6594 Views