Head to savemyexams.com for more awesome resources 5.1 Further Integration Question Paper Course Edexcel International A Level Maths Section 5. Integration Topic 5.1 Further Integration Difficulty Very Hard Time allowed: 80 Score: /60 Percentage: /100 Page 1 of 9 © 2015-2023 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers Head to savemyexams.com for more awesome resources Question 1a Find an expression for y given that dy = 5 cos 2 4x sin 4x dx [3 marks] Question 1b Integrate ∫3x (5x2 + 4) 4 dx [3 marks] Question 2 Find ∫ (2 tan x + 3 sec x) 2 dx [5 marks] Page 2 of 9 © 2015-2023 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers Head to savemyexams.com for more awesome resources Question 3a Show that 1 ∫ tan kx dx = k ln| sec kx | + c where k is a constant, and c is the constant of integration. [3 marks] Question 3b Use calculus to find the exact value of ∫ π 9 π cosec2 3θ dθ 3 cot 3θ 18 writing your answer in the form a ln b , where a and b are rational numbers to be found. [5 marks] Page 3 of 9 © 2015-2023 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers Head to savemyexams.com for more awesome resources Question 4 Use calculus to find the value of ∫2 6 ⎛⎜ 1 4 3 ⎜ + + ⎝ x 2x 3x − 2 ⎞⎟ ⎟ dx ⎠ giving your answer in the form p ln p + q ln q , where p and q are prime numbers to be found. [5 marks] Question 5a Show that ⎛⎜ cos ⎝ ⎛⎜ θ + π ⎞⎟ + sin ⎛⎜ θ + π ⎞⎟ ⎞⎟ ⎛⎜sin ⎛⎜ θ + π ⎞⎟ − cos 8⎠ 8 ⎠⎠⎝ 8⎠ ⎝ ⎝ ⎝ ⎛⎜ θ + π ⎞⎟ ⎞⎟ ≡ − cos 8 ⎠⎠ ⎝ ⎛⎜ 2θ + π ⎞⎟ 4⎠ ⎝ [3 marks] Page 4 of 9 © 2015-2023 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers Head to savemyexams.com for more awesome resources Question 5b Hence, or otherwise, find the exact value of π ∫0 8 ⎛⎜⎝sin2 ⎛⎜⎝ θ + π8 ⎞⎟⎠ − cos2 ⎛⎜⎝ θ + π8 ⎞⎟⎠ ⎞⎟⎠ dθ [3 marks] Question 6a Show that ∫0 1 ⎛ e a − 1 ⎞⎟ ⎟ e ax + b dx = e b ⎜⎜ ⎝ a ⎠ where a and b are constants, and a ≠ 0 . [3 marks] Page 5 of 9 © 2015-2023 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers Head to savemyexams.com for more awesome resources Question 6b Using your working from part [a], or otherwise, evaluate ∫0 c e ax + b dx giving your answer in terms of a,b and c , where a,b and c are constants, and a ≠ 0 . [2 marks] Question 7a Show that ⎛⎜ 1 + cot ⎛⎜ 2θ + π ⎞⎟ ⎞⎟ ⎛⎜ 1 − cot ⎛⎜ 2θ + π ⎞⎟ ⎞⎟ ≡ 2 − cosec2 ⎛⎜ 2θ + π ⎞⎟ 4 ⎠⎠⎝ 4 ⎠⎠ 4⎠ ⎝ ⎝ ⎝ ⎝ [3 marks] Question 7b Hence, or otherwise, find an expression for f (θ ) given that π π f '(θ ) = ⎛⎜ 2 + 2 cot ⎛⎜ 2θ + ⎞⎟ ⎞⎟ ⎛⎜ 2 − 2 cot ⎛⎜ 2θ + ⎞⎟ ⎞⎟ 4 4 ⎠⎠ ⎝ ⎝ ⎠⎠⎝ ⎝ [3 marks] Page 6 of 9 © 2015-2023 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers Head to savemyexams.com for more awesome resources Question 8 Show that there are no positive values of a and b that satisfy the equation ∫0 1 2a dx = ax + b ∫b 2a 1 dx x [7 marks] Page 7 of 9 © 2015-2023 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers Head to savemyexams.com for more awesome resources Question 9 Given that f '(θ) = 9 π 3 − 3 sin2 ⎛⎜ 3θ − ⎞⎟ 6 ⎠ ⎝ and also that ⎡⎢ ⎛ π ⎢⎢f ⎜ ⎢⎣ ⎝ 6 ⎞⎟ ⎤⎥⎥ 2 − ⎡⎢⎢f ⎛⎜ π ⎥ ⎢ ⎠ ⎥⎦ ⎢⎣ ⎝ 9 ⎞⎟ ⎤⎥⎥ 2 = 8 (1 − 3 ) ⎥ 3 ⎠ ⎥⎦ find f (θ ) . [6 marks] Question 10 Given that f'(x ) = f (x ) = In 1 ⎛x ⎞ ⎛ 2π ⎞⎟ 1 ⎟ = In 6 , show that tan ⎜⎜ ⎟⎟ and f ⎜⎜ 4 ⎝2⎠ ⎝ 3 ⎠ 2 3 sec ⎛⎜⎜ x ⎝2 ⎞⎟ ⎟ ⎠ [6 marks] Page 8 of 9 © 2015-2023 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers Head to savemyexams.com for more awesome resources Page 9 of 9 © 2015-2023 Save My Exams, Ltd. · Revision Notes, Topic Questions, Past Papers
