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secondary 4 | A Maths
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Is the answer 4(1+x^1/2)^3/2 / 3 + c?
Date Posted: 3 years ago
Views: 296
No.
Yup I can't solve it. I need help with it
What methods have you learnt for integration? The working will vary based on the method used.
I've only learnt integration by substitution
Pls show full question,
maybe the previous part of question asked to differentiate something that will help to integrate this. If is substitution, question will give sub equation, you need not have to come out with one yourself...
maybe the previous part of question asked to differentiate something that will help to integrate this. If is substitution, question will give sub equation, you need not have to come out with one yourself...
I don't think he's from JC or A level system, so that part on substitution equation being provided might not apply
See 2 Answers
done
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clear
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Suggested answer. Used multiple integration by parts
Date Posted:
3 years ago
Final answer is incorrect. Some errors in simplifying when going from the fourth last line to the third last line
Answer should be :
4/3 x (√x + 1)³/² - 16/15 √x (√x + 1)⁵/² + 32/105 (√x + 1)⁷/² + c
Answer should be :
4/3 x (√x + 1)³/² - 16/15 √x (√x + 1)⁵/² + 32/105 (√x + 1)⁷/² + c
Integration by substitution (twice)
① let y = √x
Then y² = x → 2y = dx/dy →"dx = 2y dy"
∫ √x √(1 + √x) dx = ∫ y √(1 + y) 2y dy = 2 ∫ y² √(1 + y) dy
② let u = √(1 + y)
Then u² = 1 + y → u² - 1 = y → 2u = dy/du →"dy = 2u du"
Then 2 ∫ y² √(1 + y) dy = 2 ∫ (u² - 1)² u (2u) du
= 4 ∫ (u⁴ - 2u² + 1)u² du
= 4 ∫ (u⁶ - 2u⁴ + u²) du
= 4 (1/7 u⁷ - 2/5 u⁵ + ⅓ u³)
= 4/7 (1 + y)⁷/² - 8/5 (1 + y)⁵/² + 4/3 (1 + y)³/²
= 4/7 (1 + √x)⁷/² - 8/5 (1 + √x)⁵/² + 4/3 (1 + √x)³/² + c
You can choose to factorise or rearrange the terms from here.
① let y = √x
Then y² = x → 2y = dx/dy →"dx = 2y dy"
∫ √x √(1 + √x) dx = ∫ y √(1 + y) 2y dy = 2 ∫ y² √(1 + y) dy
② let u = √(1 + y)
Then u² = 1 + y → u² - 1 = y → 2u = dy/du →"dy = 2u du"
Then 2 ∫ y² √(1 + y) dy = 2 ∫ (u² - 1)² u (2u) du
= 4 ∫ (u⁴ - 2u² + 1)u² du
= 4 ∫ (u⁶ - 2u⁴ + u²) du
= 4 (1/7 u⁷ - 2/5 u⁵ + ⅓ u³)
= 4/7 (1 + y)⁷/² - 8/5 (1 + y)⁵/² + 4/3 (1 + y)³/²
= 4/7 (1 + √x)⁷/² - 8/5 (1 + √x)⁵/² + 4/3 (1 + √x)³/² + c
You can choose to factorise or rearrange the terms from here.
These are some of the alternative expressions that are equivalent when you rearrange :
4/3 x (√x + 1)³/² - 16/15 √x (√x + 1)⁵/² + 32/105 (√x + 1)⁷/² + c
4/105 (√x + 1)³/² (15x - 12 √x + 8) + c
4/3 x (√x + 1)³/² - 16/15 √x (√x + 1)⁵/² + 32/105 (√x + 1)⁷/² + c
4/105 (√x + 1)³/² (15x - 12 √x + 8) + c
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