Use the trapezium rule with 6 intervals to estimate the value of ∫ 1 4 π 7 4 π cosec 1 2 x d x giving your answer correct to 2 decimal places.[Use, cosec 22 . 5 ° = 2 . 6131259297528 and also use, π = 3 . 14 , 2 = 1 . 414 ]

Given that, we have: ∫ 1 4 π 7 4 π cosec 1 2 x d x Let y = cosec 1 2 x x π 4 2 π 4 = π 2 3 π 4 4 π 4 = π 5 π 4 6 π 4 = 3 π 2 7 π 4 y 2 . 6131259297528 y 0 2 y 1 1 . 0823922002924 y 2 1 y 3 1 . 0823922002924 y 4 2 y 5 2 . 6131259297528 y 6 Now, using The trapezium rule is ∫ a b f x d x ≈ h 2 y 0 + y n + 2 y 1 + y 2 + y 3 + … + y n - 1 Here, a = 1 4 π , b = 7 4 π and width h = b - a n = 7 4 π - 1 4 π 6 = π 4 Therefore, the required area ≈ h 2 y 0 + y 6 + 2 y 1 + y 2 + y 3 + y 4 + y 5 ≈ π 4 2 2 . 6131259297528 + 2 . 6131259297528 + 2 2 + 1 . 0823922002924 + 1 + 1 . 0823922002924 + 2 ≈ π 8 2 . 6131259297528 + 2 . 6131259297528 + 2 1 . 414 + 1 . 0823922002924 + 1 + 1 . 0823922002924 + 1 . 414 ≈ π 8 × 17 . 2118207 ≈ 6 . 75563962 ≈ 6 . 75 [To 2 decimal places] Hence, the required value is 6 . 75 .

Show that ∫ 1 6 π 1 3 π 1 + 2 tan 2 x d x = 4 3 3 - π 6 .

We need to show that, ∫ 1 6 π 1 3 π 1 + 2 tan 2 x d x = 4 3 3 - π 6 Consider I = ∫ 1 6 π 1 3 π 1 + 2 tan 2 x d x ⇒ I = ∫ 1 6 π 1 3 π 1 + 2 sec 2 x - 1 d x [Using sec 2 x - tan 2 x = 1 ] ⇒ I = ∫ 1 6 π 1 3 π 1 + 2 sec 2 x - 2 d x ⇒ I = ∫ 1 6 π 1 3 π 2 sec 2 x - 1 d x ⇒ I = 2 tan x - x π 6 π 3 ⇒ I = 2 tan π 3 - π 3 - 2 tan π 6 - π 6 ⇒ I = 2 3 - π 3 - 2 3 - π 6 ⇒ I = 4 3 3 - π 6 Hence, ∫ 1 6 π 1 3 π 1 + 2 tan 2 x d x = 4 3 3 - π 6 .

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Use the trapezium rule with $2$ intervals to estimate the value of the following definite integral. Give your answers correct to $2$ decimal places.

$\int_{0}^{2} |e^{x} - 12| d x$ .[Use $e = 2.7$ ]

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Important Questions on Integration

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AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5E>Q 1

Use the trapezium rule with $2$ intervals to estimate the value of the following definite integral. Give your answers correct to $2$ decimal places.

$\int_{0}^{4} \frac{2}{3 + e^{x}} d x$ .[Use $e = 2.7$ ]

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AS and A Level

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Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5E>Q 1

Use the trapezium rule with $2$ intervals to estimate the value of the following definite integral. Give your answers correct to $2$ decimal places.

$\int_{1}^{7} \frac{2}{\ln (x + 5)} d x$ .[Use $\ln 6 = 1.79175947, \ln 9 = 2.19722458, \ln 12 = 2.48490665$ ]

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AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5E>Q 1

Use the trapezium rule with $2$ intervals to estimate the value of the following definite integral. Give your answers correct to $2$ decimal places.

$\int_{0}^{\frac{1}{3} π} \frac{1}{2 + \tan x} d x$ .[Use $\tan (\frac{π}{6}) = \frac{1}{\sqrt{3}}, \tan (\frac{π}{3}) = \sqrt{3}$ and also use $π = 3.14$ ]

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AS and A Level

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Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5E>Q 1

Use the trapezium rule with $2$ intervals to estimate the value of the following definite integral. Give your answers correct to $2$ decimal places.

$\int_{2}^{12} (\log_{10} x) d x$ . [Use, $\log_{10} 2 = 0.301, \log_{10} 7 = 0.8450, \log_{10} 12 = 1.079181246$ ]

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AS and A Level

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Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5E>Q 2

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The diagram shows part of the curve $y = x^{2} e^{- x}$ . Use the trapezium rule with $4$ intervals to estimate the value of $\int_{1}^{5} x^{2} e^{- x} d x$ , giving your answer correct to $2$ decimal places. [Use, $e = 2.7$ ]

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AS and A Level

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Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5E>Q 3

Use the trapezium rule with $6$ intervals to estimate the value of $\int_{\frac{1}{4} π}^{\frac{7}{4} π} cosec (\frac{1}{2} x) d x$ giving your answer correct to $2$ decimal places.[Use, $cosec (22.5 °) = 2.6131259297528$ and also use, $π = 3.14, \sqrt{2} = 1.414$ ]

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Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>END-OF-CHAPTER REVIEW EXERCISE 5>Q 1

Show that $\int_{\frac{1}{6} π}^{\frac{1}{3} π} (1 + 2 \tan^{2} x) d x = \frac{4 \sqrt{3}}{3} - \frac{π}{6}$ .

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AS and A Level

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Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>END-OF-CHAPTER REVIEW EXERCISE 5>Q 2

If the exact value of $\int_{- 4}^{- 2} (\frac{4}{1 - 2 x} - 2 x) d x$ in the form $a + \ln (\frac{b}{c})$ , where $a, b and$ c are integers, find the value of $a + b + c$ .

Use the trapezium rule with 2 intervals to estimate the value of the following definite integral. Give your answers correct to 2 decimal places. ∫02ex-12dx.[Use e=2.7]

Important Questions on Integration

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Use the trapezium rule with $2$ intervals to estimate the value of the following definite integral. Give your answers correct to $2$ decimal places.

$\int_{0}^{2} |e^{x} - 12| d x$ .[Use $e = 2.7$ ]