# IB Math Analysis & Approaches Question bank-Topic: SL 5.5 integration as anti-differentiation SL Paper 1

## Question

FindÂ  $$\int {\frac{1}{{2x + 3}}} {\rm{d}}x$$ .

[2]
a.

Given that $$\int_0^3 {\frac{1}{{2x + 3}}} {\rm{d}}x = \ln \sqrt P$$ , find the value of P.

[4]
b.

## Markscheme

$$\int {\frac{1}{{2x + 3}}} {\rm{d}}x = \frac{1}{2}\ln (2x + 3) + C$$Â  (accept $$\frac{1}{2}\ln |(2x + 3)| + C$$Â )Â Â  Â A1A1Â Â Â Â  N2

[2 marks]

a.

$$\int_0^3 {\frac{1}{{2x + 3}}} {\rm{d}}x = \left[ {\frac{1}{2}\ln (2x + 3)} \right]_0^3$$

evidence of substitution of limitsÂ Â Â Â  (M1)

e.g.$$\frac{1}{2}\ln 9 – \frac{1}{2}\ln 3$$

evidence of correctly using $$\ln a – \ln b = \ln \frac{a}{b}$$Â (seen anywhere)Â Â Â Â  (A1)

e.g. $$\frac{1}{2}\ln 3$$

evidence of correctly using $$a\ln b = \ln {b^a}$$ (seen anywhere)Â Â Â Â  (A1)

e.g. $$\ln \sqrt {\frac{9}{3}}$$

$$P = 3$$ (accept $$\ln \sqrt 3$$ )Â Â Â Â  A1 Â  Â  N2

[4 marks]

b.

## Question

A function fâ€‰(x) has derivative fâ€‰â€²(x) = 3x2 + 18x. The graph of f has an x-intercept at x = âˆ’1.

FindÂ fâ€‰(x).

[6]
a.

The graph of f has a point of inflexion at x = p. Find p.

[4]
b.

Find the values of x for which the graph of f is concave-down.

[3]
c.

## Markscheme

evidence of integrationÂ  Â  Â  Â (M1)

egÂ Â $$\int {f’\left( x \right)}$$

correct integration (accept absence of C)Â  Â  Â  Â (A1)(A1)

egÂ  $${x^3} + \frac{{18}}{2}{x^2} + C,\,\,{x^3} + 9{x^2}$$

attempt to substitute x = âˆ’1 into their fÂ = 0 (must have C)Â  Â  Â  M1

egÂ Â $${\left( { – 1} \right)^3} + 9{\left( { – 1} \right)^2} + C = 0,\,\, – 1 + 9 + C = 0$$

Note: Award M0 if they substitute into original or differentiated function.

correct workingÂ  Â  Â  Â (A1)

egÂ Â $$8 + C = 0,\,\,\,C =Â – 8$$

$$f\left( x \right) = {x^3} + 9{x^2} – 8$$Â  Â  Â Â A1 N5

[6 marks]

a.

METHOD 1 (using 2nd derivative)

recognizing that f” = 0 (seen anywhere)Â  Â  Â  M1

correct expression forÂ f”Â  Â  Â  (A1)

egÂ  Â 6x + 18, 6p + 18

correct workingÂ  Â  Â  (A1)

6pÂ + 18 = 0

pÂ =Â âˆ’3Â  Â  Â  Â A1 N3

METHOD 1Â (usingÂ 1st derivative)

recognizing the vertex of fâ€² is neededÂ  Â  Â  Â (M2)

egÂ  Â $$– \frac{b}{{2a}}$$Â (must be clear this is for fâ€²)

correct substitutionÂ  Â  Â Â (A1)

egÂ  Â $$\frac{{ – 18}}{{2 \times 3}}$$

pÂ =Â âˆ’3Â  Â  Â  Â A1 N3

[4 marks]

b.

valid attempt to use f”â€‰(x) to determine concavityÂ  Â  Â  (M1)

egÂ Â Â f”â€‰(x) < 0,Â f”â€‰(âˆ’2),Â f”â€‰(âˆ’4),Â  6x + 18 â‰¤ 0Â

correct workingÂ  Â  Â  Â (A1)

egÂ  Â 6xÂ + 18Â < 0,Â f”â€‰(âˆ’2) = 6,Â f”â€‰(âˆ’4) =Â âˆ’6Â

f concave down for x < âˆ’3 (do not accept xÂ â‰¤ âˆ’3)Â  Â  Â  Â A1 N2

[3 marks]

c.

## Question

LetÂ $$f\left( x \right) = 6{x^2} – 3x$$.Â The graph ofÂ $$f$$Â is shown in the following diagram.

Find $$\int {\left( {6{x^2} – 3x} \right){\text{d}}x}$$.

[2]
a.

Find the area of the region enclosed by the graph ofÂ $$f$$,Â the x-axis and the linesÂ x = 1 and x = 2 .

[4]
b.

## Markscheme

$$2{x^3} – \frac{{3{x^2}}}{2} + c\,\,\,\left( {{\text{accept}}\,\,\frac{{6{x^3}}}{3} – \frac{{3{x^2}}}{2} + c} \right)$$Â  Â  Â A1A1 N2

Notes: Award A1A0 for both correct terms if +c is omitted.
Award A1A0 for one correct term eg $$2{x^3} + c$$.
Award A1A0 if both terms are correct, but candidate attempts further working to solve for c.

[2 marks]

a.

substitution of limits or function (A1)

egÂ Â $$\int_1^2 {f\left( x \right)} \,{\text{d}}x,\,\,\left[ {2{x^3} – \frac{{3{x^2}}}{2}} \right]_1^2$$

substituting limits into their integrated function and subtractingÂ  Â  Â (M1)

egÂ Â $$\frac{{6 \times {2^3}}}{3} – \frac{{3 \times {2^2}}}{2} – \left( {\frac{{6 \times {1^3}}}{3} + \frac{{3 \times {1^2}}}{2}} \right)$$

Note: Award M0 if substituted into original function.

correct workingÂ  Â  Â  (A1)

egÂ Â $$\frac{{6 \times 8}}{3} – \frac{{3 \times 4}}{2} – \frac{{6 \times 1}}{3} + \frac{{3 \times 1}}{2},\,\,\left( {16 – 6} \right) – \left( {2 – \frac{3}{2}} \right)$$

$$\frac{{19}}{2}$$Â  Â  Â A1 N3

[4 marks]

b.