1.11.5 Integration by substitution and by parts (A-level only)

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Question 92

Determine the indefinite integral $dx\int \frac{\ln(x^2)}{x^3} \, \text{d}x$ .

[3]

A mechanical piston is subject to a variable resistive force $F (x)$ , where $x$ is the displacement in metres from the start of the stroke. The force, in Newtons, is given by

\frac{3 + 2x^2 + \ln(x^2)}{x^3}, \quad x \ge 1

The work done by the force as the piston moves from $x = 1$ to $x = 2$ is given by $dx\int_1^2 F(x) \, \text{d}x$ .

Using the result from part (a), find the exact work done, writing your answer in the form $\ln b$ , where $a$ and $b$ are constants to be determined.

[5]

Determine the indefinite integral $dx\int \frac{\ln(x^2)}{x^3} \, \text{d}x$ .

[3]

A mechanical piston is subject to a variable resistive force $F (x)$ , where $x$ is the displacement in metres from the start of the stroke. The force, in Newtons, is given by

\frac{3 + 2x^2 + \ln(x^2)}{x^3}, \quad x \ge 1

The work done by the force as the piston moves from $x = 1$ to $x = 2$ is given by $dx\int_1^2 F(x) \, \text{d}x$ .

Using the result from part (a), find the exact work done, writing your answer in the form $\ln b$ , where $a$ and $b$ are constants to be determined.

1.11.5 Integration by substitution and by parts (A-level only)

Question 92

Markscheme

1.11.5 Integration by substitution and by parts (A-level only)

Question 92

Markscheme

1.11.5 Integration by substitution and by parts (A-level only)

Question 92

1.11.5 Integration by substitution and by parts (A-level only) Questions

1.11.5 Integration by substitution and by parts (A-level only)

Question 92

1.11.5 Integration by substitution and by parts (A-level only) Questions