RELIABLE CLASSES / S.Y.J.C./ MATHEMATICS AND STATISTICS-I
CHAPTER - 6 INTEGRATION
Marks allotted : 08
Definition of integral of a function:
If f (x) and g (x) are two functions such that then f (x) is called an integral or an antiderivative or a primitive of g (x) with respect to x. It is denoted by g (x) dx = f (x) and read as integral of g (x) w.r.t.x. is f (x). Here, we say that g (x) is an integrand.
The process of finding the primitive or integral of a function is called integration. Thus, integration is the inverse process of a differentiation.
Functions / Integrations / Reasons1. 0 / C /
2. 1 / /
3. xn (n ≠ -1) / /
4. / log (x) /
5. / /
6. / /
7. a(a > 0) / /
8. sinx / - cosx /
9. cos x / Sinx /
10. Sec2 x / tan x /
11. Sec x tan x / Sec x /
12. Cosec2 x / - Cotx /
13. Cosec Cotx / - Cosec x /
14. / tan-1 (x) /
OR / OR
- cot-1 (x) /
15. / Sin-1 (x) /
OR / OR
- Cos-1 (x) /
16. / Sec-1 (x) /
OR / OR
Cosec-1 (x) / =
PART (A)
Type 1. Using Direct Method.
Q1. (1 + 2x + 3x2 + 4x3)dxQ2.
Q3.Q4.(5 – 3x) (2 + x2) dx
Q5.If f (x) = 6x2 + 4x + 5 and f (0) = 5, find f (x).
Q6.If f (x) = 8x3 + 3x2 – 10x – k and f (0) = -3 and f (-1) = 0, find f (x).
Q7.If f (x) = 4x3 – 3x2 + 2x + k and f (0) = 1; f (1) = 4, find f (x)
Q8.If f (x) = k (cos x – sin x), f(0) = 3 and = 15, find f (x).
Type 2: Composite Function
1. / / (ax + b)n (n -1) /2. / Log lxl / /
3. / 2 / /
4. ex / ex / eax+b /
5. ax (a > 0) / / amx+n (a > 0) /
6. Sinx / -Cosx / Sin (ax +x) /
7. Cosx / Sinx / Cos (ax + b) /
8. Sec2 x / tanx / Sec2 (ax + b) /
9. Sec x tan x / Secx / Sec (ax + b) tab (ax+ b) / sec (ax+ b)
10. Cosec2 x / -Cotx / Cosec 2(ax +b) / cot (ax+ b)
11. Cosec Cotx / -CoSecx / Cosec (ax + b) /
12. / tan-1 x / /
13. / Sin-1 (x) / /
14. / Sec-1 (x) / /
Integrate the following functions w.r.t.x where f(x) is given by
Q9.(3x2 – 5)2dxQ10.
Q11.dxQ12. a3x-1dx
Q13. cos (3-2x) dxQ14. cosec (4-3x) cot (4-3x) dx
Q15.
Type 3: Using Where n ≠ -1 put t = f(x) and solve
Q16.x2 (2x3 – 1)3/2dxQ17.dx
Q18. cos7 x sin x dxQ19. tan5x sec2 x dx
Q20. dxQ21. dxQ22.dx
Type 4 : Using Where n = -1 Put t = f(x) and solve
Find the following integrals
Q23.Q24.
Q25.Q26.
Q27.Q28.
Q29.Q30.
Q31.Q32.
Q33.Q34.
Q35.Q36
Type 5: Using Put t = f(x) & Solve
Q37.Q38.39.
Type 6:On f [g(x)] g(x) dxput t = g(x) & solve
Q40.Q41.Q42.
Q43.Q44.7xlogx (1+ log x) dxQ45.
Q46.
Type 7: p (x) (ax + b)n dx
Where p (x) is polynomial function in x and n is a rational number
Put t = ax + b
Find the value of x in terms of t &
i.e.t = ax + b,t – b = ax,
Q47.dxQ48.dx
Q49.(5 – 3x) (2-3x)-1/2dxQ50.dx
Type 8 On Trigonometric Function
1. sinx dx = - Cos x + c
2. cos x dx = Sin x + c
3. sec2xdx = tan x + c
4. tan2 x dx = tanx – x + c
5. Secx tanx dx = sec x + c
6. Cosec2xdx = -Cotx + c
7. Cot2xdx = - Cotx – x + c
8. Cosec x cotx dx = - Cosec x + c
9. Cotx dx = log
10. tanx dx = log
11. sec x dx = log OR log + c
12. Cosecxdx = log OR log
Integrate the following w.r.t.x
Q51.(i)(ii)(iii)sec2 x – sec x tan x dx
Q52.(i)(ii)
Q53.(i)sin 2x cos 8x(ii)
Q54.(i) Sin2xdx(ii)Cos2xdx
Q55.(i) Sin4xdx(ii)Cos4xdx
Q56.(i)Sin3xdx(ii)Cos3xdx
Q57.Q58.where A is constant.
Q59.Q60.
Type 9: On Compound Angle.
Write the angle of Numerator same as denominator without changing the original angle ofnumerator, Apply Compound angle formula in numerator Simplify and integrate:
Q61.Q62.
Q63.Q64.
Note.
1.If the terms in the denominator are similar i.e. sin (x – a). sin (x – b) or cos (x – a) cos (x – b), we introduce sin (b-a) which gives dissimilar terms.
2.If the terms in the denominator are dissimilar, i.e. sin (x – a) cos (x – b) or cos (x –a) sin (x – b), we introduce cos (b – a) which give similar terms.
Q65.
Q66. dx
Q67.dx
Part B
Type 1: On Standard integration formula
Integrals of the type
In order to find this type of integrals we may use the following steps.
Step 1: Make the coefficient of x2 unity, if it is not, as
Step 2: Add and subtract the square of the half of coefficient of x i.e. to complete the square
=
Step 3: Use the suitable formula for evaluation.
1.
2.
3.
Q68. Integrate the following.
(i)(ii)
(iii)(iv)
(v)
Q69.(i)(ii)
(iii)(iv)
(v)(vi)
(vii)(viii)
(ix)(x)
Q70.(i)(ii)
(iii)(iv)
(v)(vi)
(vii)
Type 2
4.
5.
6.
Q71.(i)(ii)
(iii)(iv)
Q72.(i)(ii)
(iii)(iv)
(v)(vi)
(vii)(viii)
(ix)(x)
Q73.
(i)(ii)
(iii)(iv)
(v)(vi)
(vii)(viii)
(ix)
Type 3:
7.
8.
9.
Q74.Find the following integrals
(i)(ii)
(iii)(iv)
(v)(vi)
(vii)(viii)
(ix)(x)
Type 4:
dx
Divide each term by Cos2x if in denominator sec2x = 1 + tan2x put t = tan x and solve
Q75.Q76.
Q77.(i) (ii)
Type 5:
Put t = tanif the given angle in the question is x and put t = tan x if the angle in the question is 2x and solve.
Put t = tan
Sin x =
Integrate the following w.r.t.x
Q78.Q79.
Q80.Q81.
Q82.Q83.
Q84.Q85.
Type 6:
On Rational Algebraic Functions.
(A)Degree of Numerator < Degree of Denominator
Write the derivative of denominator in numerator without Changing the original value of numerator, simplify and apply.
When than first part will be always
Q86Find the following integrals
(i).(ii).(iii).
When than first part will always
Q87.(i)(ii)
(iii)(iv)
(v)
(B)Degree of Numerator = Degree of Denominator
Write numerator same as denominator without changing the original value of numerator, simplify and integrate.
When then first part will be always
Q88.Integrate the following w.r.t.x
(i)(ii)(iii)
When then first part will be always
Q89.(i)(ii)(iii)
(c)Degree of Numerator < Degree of Denominator
Apply algebraic division and use the property
Dividend = Division x Quotient + Remainder
Q90.(i)(ii)(iii)
PART –C
Type 1: Integration by Partial Fractions
The following table indicates how to choose the partial fractions corresponding to the given rational function in different forms.
Sr. No. / Form of the rational function / Form of the partial fraction1. / /
2. / are distinct. /
3. / /
4. / /
5. / /
6. /
/
Q91.Find the following integrals
(i)(ii)
(iii)(iv)
(v)(vi)
(vii)
Q92.(i)(ii)(iii)
Q93.(i)(ii)
Q94.Q95.
Type 2: Using
Q96.Q97.
Q98.Q99.
Q100.Q101.
Q102.Q103.
Q104.Q105.
Q106.Q107.Q108. [ sin (log x) + (cos (log x)]dx
Type 3 : Integration by parts
First decide u & v by LIATE rule than integration
L =Logarithmic function Eg. logx
I =Inverse Trigonometric Function E.g. Sin-1 (x)
A =Algebraic Function Eg. x, x2.
T =Trigonometric function Eg. Sinx, Cosx
E =Exponential Function Eg. ex, ax
Q109Find the following integrals”
(i)(ii)
Q110.(i)(ii)
Q111.(i)(ii)
Q112.Q113.Q114.
Q115.Q116.Q117.
Q118.Q119.Q120
1