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Integral of (2(x^3)-7)/((x^4)-14x)

Set f(x)= (x^4)-14x. f’(x)=4(x^3)-14=2(2(x^3)-7). Thus we can write (2(x^3)-7)/((x^4)-14x)=(1/2)f’(x)/f(x). The integral of f’(x)/f(x)=ln|f(x)|+c. Thus the integral of (2(x^3)-7)/((x^4)-14x) is (1/2)(ln|f(x)...
IK
Answered by Issy K. • Maths tutor
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A curve has parametric equations: x = 3t +8, y = t^3 - 5t^2 + 7t. Find the co-ordinates of the stationary points.

First differentiate: dx/dt = 3, dy/dt = 3t 2 - 10t + 7 Using the chain rule: dy/dx = dy/dt * dt/dx = (3t 2 - 10t + 7)/3 At stationary points, the gradient is equal to zero: 3t 2 - 10t + 7 = 0 Solve for t usi...
RB
Answered by Robbie B. • Maths tutor
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Integrate tan (x) with respect to x.

I = ∫ Tan (x) dx= ∫ (sin(x)) / (cos(x)) dx We see that this is close to the standard integral ∫ F'(x) / F(x) dx = Ln (F(x)) + C So first we must rewrite the Integral as: I = - ∫ (-sin(x)) / (cos(x)) dx (Taki...
MH
Answered by Matthew H. • Maths tutor
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Find the tangent to y = x^2 - 4x + 9 at the point (3,15)

First find dy/dx: dy/dx = 4x - 4 And thus at (3,15): dy/dx = 12 - 4 = 8 = m (as m is the gradient of a curve) So using y - y 1 = m(x - x 1 ) where (x 1 ,y 1 ) = (3,15): y - 15 = 8(x - 3) y = 8x- 9
SH
Answered by Scott H. • Maths tutor
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Find the derivative (dy/dx) of the curve equation x^2 -y^2 +y = 1.

Most of the differentiation problems require us to apply one of the well known rules, be it product rule, quotient rule or chain rule. But those problems have one thing in common: explicite formula for y, be...
AG
Answered by Adam G. • Maths tutor
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