Integration by substitution

Use the substitution t tan to show that

Hence show that

where A and B are constants to be determined and k is an arbitrary constant.

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

Use the substitutiont = tan to show that

where a, b and c are constants to be determined.

Hence show that

Show that the substitution t tan transforms the integral

into the integral

Hence determine

Use a hyperbolic substitution and calculus to show that

where k is an arbitrary constant.

Figure 1 shows a sketch of part of the curve C with equation

The finite region R, shown shaded in Figure 1, is bounded by the curve C, the x‑axis and the line with equation x = 3

Using algebraic integration and the result from part (a), show that the area of R is given by

Use a hyperbolic substitution and calculus to show that

where k is an arbitrary constant.

Figure 1 shows a sketch of part of the curve C with equation

The finite region R, shown shaded in Figure 1, is bounded by the curve C, the x‑axis and the line with equation x = 3

Using algebraic integration and the result from part (a), show that the area of R is given by

Use the substitution t = tan to prove the identity

Use the substitution t = tan to determine the exact value of

giving your answer in simplest form.

show that f (x) can be written in the form

Hence solve, for 0 x 2π, the equation

giving your answers to 2 decimal places where appropriate.

Use the result of part (a) to show that

where K is a constant to be determined.

where k is an arbitrary constant.