Indefinite integration

Find

giving the answer in simplest form.

Find

writing each term in simplest form.

The curve C has equation y = f (x) The curve

• passes through the point P (3, –10)

• has a turning point at P

Given that

where k is a constant,

show that k = 12

Hence find the coordinates of the point where C crosses the y-axis.

Figure 2 shows a sketch of part of the curve with equation y = x (x + 2)(x – 4).

The region R₁ shown shaded in Figure 2 is bounded by the curve and the negative x-axis.

Show that the exact area of R₁ is

The region R₂ also shown shaded in Figure 2 is bounded by the curve, the positive x-axis and the line with equation x = b, where b is a positive constant and 0 b 4

Given that the area of R₁ is equal to the area of R₂

verify that b satisfies the equation

The roots of the equation 3b² – 20b + 20 = 0 are 1.225 and 5.442 to 3 decimal places. The value of b is therefore 1.225 to 3 decimal places.

Explain, with the aid of a diagram, the significance of the root 5.442

Given that k ⁺

Find, in its simplest form,

Hence find

Use integration by parts to find

giving your answer in the form a + b ln 2, where a and b are constants to be found.

Find

Find

Given that y = at x = 0, solve the differential equation

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Given that y = 8 at x =1, solve the differential equation

giving your answer in the form y² = g(x).

Use the substitution u^x to show that = 1 + e

where k is a constant.

Given that y = 4x³ – , x  0, find in their simplest form

A curve with equation y = f(x) passes through the point (4, 9). Given that

find f(x), giving each term in its simplest form.

Point P lies on the curve . The normal to the curve at P is parallel to the line 2 y + x = 0

Find the x coordinate of P.

Given that y = 2x⁵ x 0, find in their simplest form

Given that y = 37 at x = 4, find y in terms of x, giving each term in its simplest form.

Find

giving each term in its simplest form.

Find

giving each term in its simplest form. (4)

Show that f'(x) = 9x^–2 + A + Bx², where A and B are constants to be found.

Find f " (x).

Given that the point (–3, 10) lies on the curve with equation y = f (x),

find f (x).

Given that

equal to 10 (4)

Given that y = 7 at x = 1, find y in terms of x, giving each term in its simplest form.

Find

giving each term in its simplest form.

The point P (4, –1) lies on the curve C with equation y = f(x), x > 0, and

Find the equation of the tangent to C at the point P, giving your answer in the form y = mx + c, where m and c are integers.

Find f(x).

Find

giving each term in its simplest form.

The curve C has equation y = f(x), x > 0, where

Given that the point P (4, 5) lies on C, find

f(x),

an equation of the tangent to C at the point P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.

Given that y = 35 at x = 4, find y in terms of x, giving each term in its simplest form.

Find, giving each term in its simplest form. 12x⁵ 8x³+3dx

A curve has equation y = f(x) and passes through the point (4, 22). Given that

use integration to find f(x), giving each term in its simplest form.

The gradient of a curve C is given by = x 0.

The point (3, 20) lies on C.

Find an equation for the curve C in the form y = f(x).

Find (3² 4⁵ 7) d.

Given that the point P (4, 1) lies on C,

find f(x) and simplify your answer.

Find an equation of the normal to C at the point P (4, 1).