Integrating hyperbolic & inverse functions

Express

in partial fractions.

Hence, show that

Express as partial fractions

Hence, show that

where a and b are constants to be determined.

Express as partial fractions

Hence, show that

where a and b are constants to be determined.

Explain why ₀cosh x dx is an improper integral.

Show that ₀cosh x dx is divergent.

4 sinhx = pcoshxwhere p is a real constant

Given that this equation has real solutions, determine the range of possible values for p

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

Solutions based entirely on graphical or numerical methods are not acceptable.

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

y = arsinhx x 0

and the straight line with equation y = β The line and the curve intersect at the point with coordinates (α, β)

The finite region R, shown shaded in Figure 1, is bounded by the curve with equation y = arsinh x, the y-axis and the line with equation y = β

The region R is rotated through 2π radians about the y-axis.

Use calculus to find the exact value of the volume of the solid generated.

Solutions based entirely on graphical or numerical methods are not acceptable.

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

y = arsinhx
x 0

and the straight line with equation y = β The line and the curve intersect at the point with coordinates (α, β)

The finite region R, shown shaded in Figure 1, is bounded by the curve with equation y = arsinh x, the y-axis and the line with equation y = β

The region R is rotated through 2π radians about the y-axis.

Use calculus to find the exact value of the volume of the solid generated.

Show that, for n 0

Find I₂