Let I = ∫ 6 x d x Using: ∫ k f x d x = k ∫ f x d x , k ∈ R ⇒ I = 6 ∫ 1 x d x Using: ∫ 1 a x + b d x = 1 a ln ( a x + b ) + c , a x + b > 0 Therefore, I = 6 ln x + c , where c = constant of integration.

Find: ∫ 1 2 x d x .

Let I = ∫ 1 2 x d x Using: ∫ k f x d x = k ∫ f x d x , k ∈ R ⇒ I = 1 2 ∫ 1 x d x Using: ∫ 1 a x + b d x = 1 a ln ( a x + b ) + c , a x + b > 0 Therefore, I = 1 2 ln x + c , where c = constant of integration.

Find: ∫ 1 3 x + 1 d x .

Let I = ∫ 1 3 x + 1 d x It is of the form of ∫ 1 a x + b d x Using: ∫ 1 a x + b d x = 1 a ln ( a x + b ) + c , a x + b > 0 Therefore, I = 1 3 ln ( 3 x + 1 ) + c , valid for 3 x + 1 > 0 where c = constant of integration.

Find: ∫ 6 2 x - 5 d x .

Let I = ∫ 6 2 x - 5 d x ⇒ I = 6 ∫ 1 2 x - 5 d x Using: ∫ 1 a x + b d x = 1 a ln ( a x + b ) + c , a x + b > 0 Therefore, I = 6 3 ln ( 2 x - 5 ) + c , valid for 2 x - 5 > 0 ⇒ I = 3 ln ( 2 x - 5 ) + c , where c = constant of integration. Hence, ∫ 6 2 x - 5 d x = 3 ln ( 2 x - 5 ) + c .

Find: ∫ 5 2 - 3 x d x .

Let I = ∫ 5 2 - 3 x d x ⇒ I = 5 ∫ 1 2 - 3 x d x Using: ∫ 1 a x + b d x = 1 a ln ( a x + b ) + c , a x + b > 0 Therefore, I = 5 - 3 ln ( 2 - 3 x ) + c , valid for 2 - 3 x > 0 ⇒ I = - 5 3 ln ( 2 - 3 x ) + c , where c = constant of integration. Hence, ∫ 5 2 - 3 x d x = - 5 3 ln ( 2 - 3 x ) + c .

Find: ∫ 3 2 ( 5 x - 1 ) d x .

Let I = ∫ 3 2 ( 5 x - 1 ) d x ⇒ I = 3 2 ∫ 1 ( 5 x - 1 ) d x Using: ∫ 1 a x + b d x = 1 a ln ( a x + b ) + c , a x + b > 0 Therefore, I = 3 2 × 1 5 ln ( 5 x - 1 ) + c , valid for ( 5 x - 1 ) > 0 ⇒ I = 3 10 ln ( 5 x - 1 ) + c , where c = constant of integration. Hence, ∫ 3 2 ( 5 x - 1 ) d x = 3 10 ln ( 5 x - 1 ) + c .

Find: ∫ 0 5 1 x + 2 d x .

Let I = ∫ 0 5 1 x + 2 d x Using: ∫ 1 a x + b d x = 1 a ln ( a x + b ) + c , a x + b > 0 Therefore, I = ln ( x + 2 ) 0 5 , valid for x + 2 > 0 ⇒ I = ln ( 5 + 2 ) - ln ( 0 + 2 ) ⇒ I = ln 7 - ln 2 ⇒ I = ln 7 2 Hence, ∫ 0 5 1 x + 2 d x = ln 7 2 .

E M B I B E

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5A>Q 11

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The diagram shows the graph of $y = e^{x}$ . The points $(\ln 2, 2) and (\ln 3, 3)$ lie on the curve. Find the value of $\int_{\ln 2}^{\ln 3} e^{x} d x$ .

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Important Questions on Integration

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 5 - Integration>EXERCISE 5A>Q 11

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The diagram shows the graph of $y = e^{x}$ . The points $(\ln 2, 2) and (\ln 3, 3)$ lie on the curve. Show that $\int_{2}^{3} \ln y d y = \ln (\frac{27}{4 e})$ .