Expression. Substitution
1) a² + x². x= a tan k or a cot k
2) a² - x². x= a sin k or a cos k
3) x² + a². x= a sec k ora cosec k
4) √{(a-x)/(a+x)} or √{(a+x)/(a- x) x = a cos 2k
5) √{(x- ¢)/(¥-x)} or √{(x-¢)(x- ¥). x = ¢ cos² k + ¥ sin² k
FORMULA;
1) dx/(x²+ a²)= 1/a tan⁻¹(x/a)
2) dx/(x²- a²)= 1/2a log|(x-a)/(x+a)|
3) dx/(a²- x²)= 1/2a log|(a+x)/(a - x)|
4) dx/√(a²- x²)= sin⁻¹(x/a)
5) dx/√(a²+ x²)= log|x + √(a² +x²)|
6) dx/√(x²- a²)= log|x + √(x² - a²)|
EXERCISE --1
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1) ∫ dx/(x²+25). 1/5 tan⁻¹(x/5)
2) ∫ dx/(4+ 9x²). 1/6 tan⁻¹(3x/2)
3) ∫ dx/(4x²+9). 1/6 tan⁻¹(2x/3)
4) ∫ dx/(9x² -1). 1/6 log|(3x-1)/(3x+1)|
5) ∫ dx/(9x² -4). 1/12 log|(3x-2)/(3x+2)|
6) ∫ dx/(16 - 9x²). 1/24 log|(4+3x)/(4 - 3x)|
7) ∫ dx/(a² - m²x²). 1/2am log|(a+ mx)/(a - mx)|
8) ∫ dx/(a²x² - b²). 1/2ab log|(ax- b)/(ax + b)|
9) ∫ dx/(a²x² + b²). 1/ab tan⁻¹(ax/b)
10) ∫ dx/√(4 + x²). log|x+ √(x²+4)|
11) ∫ dx/√(1 + 4x²). 1/2 log|2x + √(4x²+1)|
12) ∫ dx/√(9 - 25x²). 1/5 sin⁻¹(5x/3)
13) ∫ dx/√(16x²+ 25). 1/4 log|x + √(x² +25/16)
14) ∫ dx/√(4x² -9). 1/2 log(x+ √(x²- 9/4)
15) ∫ dx/√(a²+ b²x²). 1/b log|(bx+ √(a² + b²x²|
16) ∫ dx/√(a² - b²x²). 1/b sin⁻¹(bx/a)
17) ∫ dx/√((2- x)²+1). - log|(2- x) + √((2- x)² +1)|
18) ∫ dx/√((2- x)²-1). - log|(2- x) + √((2- x)² -1)|
19) ∫ dx/((x+2)²+1). tan⁻¹{(x+2)/1}
20) ∫ (x²-1)/(x²+4). x - 5/2 tan⁻¹(x/2)
21) ∫ (x⁴+1)/(x²+1). x³/3 - x + 2 tan⁻¹x
EXERCISE --2
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1) ∫ dx/(x² - x +1). 2/√3 tan⁻¹{(2x-1)/√3}
2) ∫ dx/(x² +8x +20). 1/2 tan⁻¹{(x+4)/2}
3) ∫ dx/(x² +2x +5). 1/2 tan⁻¹{(x+1)/2}
4) ∫ dx/(x² +4x +8). 1/2 tan⁻¹{(x+2)/2}
5) ∫ dx/(x² - 10x +34). 1/3 tan⁻¹{(x-5)/3}
6) ∫ 3/(x² - 8x +25). tan⁻¹{(x-4)/3}
7) ∫ dx/(2x² + x -1). 1/3 log|+2x-1)/(2(x+1)|
8) ∫ dx/(3x² +13x -10). 1/17 log|(3x-2)/3(x+5)|
9) ∫ dx/(4x² - 4x +3). 1/2√2 tan⁻¹{(2x-2)/√2
10) ∫ dx/(9x² +6x +10). 1/9 tan⁻¹{(3x+1)/3}
11) ∫ dx/(3+ 2x -x²). 1/4 log|(x+1)/(3-x)|
12) ∫ dx/(1+ x -x²). 1/√5 log|(√5 - 1 + 2x)/(√5 + 1- 2x)|
13) ∫ dx/(3- 2x -x²). 1/4 log|(3+ x)/(1 -x)|
14) ∫ dx/(7 - 6x -x²). 1/8 log|(7+x)/(1 -x)|
EXERCISE -- 3
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1) ∫ x/(x⁴+ x²+1). 1/√3 tan⁻¹{(2x²+1)/3
2) ∫ x/(x⁴+ 2x²+3). 1/2√2 tan⁻¹{(x²+1)/√2}
3) ∫ dx/{x(xⁿ+1)}. 1/n log{xⁿ/(xⁿ+1)}
4) ∫ dx/{x(x⁵+1)}. 1/5 log |x⁵/(x⁵+1)|
5) ∫ 3x⁵/(1+ x¹²)}. 1/2 tan⁻¹(x⁶)
6) ∫ x²/(x⁶ - a⁶). 1/(6a³) log|(x³ - a³)/(x³ + a³)|
7) ∫ x²/(x⁶ + a⁶). 1/(3a³) tan⁻¹(x³/a³)
8) ∫ dx/{x(x⁶ + 1)}. 1/6 log|x⁶/(x⁶+1)|
9) ∫ dx/{x(x³+ 1)}. 1/3 log|x³/(x³+1)|
10) ∫ dx/{x(x⁴+ 1)}. 1/4 log|x⁴/(x⁴+1)|
11) ∫ x/(x⁴ - x²+1). 1/√3 tan⁻¹{(2x²-1)/√3}
12) ∫ x/(3x⁴ - 18x² + 11). √3/48 log |(x² - 3 - 4/√3)/(x² - 3 + 4/√3)|
13) ∫ eˣ/(e²ˣ + 6eˣ+5). 1/4 log|(eˣ+1)/(eˣ+5)|
14) ∫ eˣ/(e²ˣ + 5eˣ+6). log|(eˣ+2)/(eˣ+3)|
15) ∫ eˣ/(1+ e²ˣ). tan⁻¹(eˣ/1)
16) ∫ e³ˣ/(4e⁶ˣ - 9). 1/36 log|(2e³ˣ- 3)/(2e³ˣ+3)|
17) ∫ e⁻ˣ/(16+ 9e⁻²ˣ). -1/12 tan⁻¹(3e⁻ˣ/4)
18) ∫ dx/(eˣ +e⁻ˣ). tan⁻¹(e⁻ˣ)
19) ∫ dx/x{(6 log x)² + 7 log x +2}. Log |(2 log x +1)/(3 log x+2)|
20) ∫ sinx/(1+ cos²x). tan⁻¹(cosx)
21) ∫ cosx/(sin²x+ 4 sinx +5). tan⁻¹(sinx+2)
21) ∫ sec²x/(1 - tan²x). 1/2 log|(1+ tanx)/(1- tanx)|
** FORMULA
1) dx/√(a² + x²)= log|x+ √(a²+x²)|
2) dx/√(x² - a²)= log|x+ √(x²- a²)|
3) dx/√(a² - x²)= sin⁻¹(x/a)
EXERCISE-4
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1) ∫dx/√(2- 4x+ x²). log|(x-2)+ √(x² - 4x +2)|
2) ∫ dx/√(x² - 4x +2). log|(x-2)+ √(x² - 4x +2)|
3) ∫ dx/√(2x- x²). sin⁻¹(x- 1)
4) ∫ dx/√(8+ 3x- x²). sin⁻¹{(2x- 3)/√41}
5) ∫ dx/√(2- 4x- x²). sin⁻¹{(x +2)/√6}
6) ∫ dx/√(5+ 4x- x²). sin⁻¹{(x- 2)/3}
7) ∫ dx/√(1+ 2x- 3x²). 1/√3 sin⁻¹{(3x- 1)/2}
8) ∫ dx/√{(2- x)²+1}. - log|2-x + √(x² - 4x +5)|
9) ∫ dx/√{(2- x)² -1}. - log|2- x + √(x² - 4x +3)|
10) ∫ dx/√(x²+ 4x+2). log|x+2 + √(x² + 4x+ 2)|
11) ∫ dx/√(8+ 4x- 4x²). 1/2 sin⁻¹{(2x- 1)/3}
12) ∫ dx/√(3x²+ 5x+7). 1/√3 log{x + 5/6 + √(x²+5x/3 +7/3}}
13) ∫ dx/√(1+ x- x²). sin⁻¹{(2x- 1)/√5}
14) ∫ dx/√(8+3x- x²). sin⁻¹{(2x- 3)/√41
15) ∫ dx/√(7- 3x- 2x²). 1/√2 sin⁻¹{(4x +3)/√65}
16) ∫ dx/√(6 - x- x²). sin⁻¹{(2x+ 1)/5}
17) ∫ dx/√(6+ x- x²). sin⁻¹{(2x- 1)/5}
18) ∫ dx/√(4- 2x- x²). sin⁻¹{(x+ 1)/√5}
19) ∫ dx/√(7- 6x- x²). sin⁻¹{(x +3)/4}
20) ∫ dx/√(2+2x- x²). sin⁻¹{(x- 1)/√3}
21) ∫ dx/√(x²+ 12x+ 11). log|x + 6 + √(x² +12x +11)|
22) ∫ dx/√{(x+5)(x+1)}. Log|(x+3) + √(x²+ 6x+ 5)|
23) ∫ dx/√{(x-1)(x-2)}. Log|(x -3)/2 + √(x²- 3x +2)|
24) ∫ dx/√{x(1- 2x)}. 1/√2 sin⁻¹{(4x -1)}
25) ∫ dx/√{(x- a)(b -x)}. 2 log|√(x- a) + √(x-b)|
EXERCISE-5
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1) ∫ x²/√(1- x⁶). 1/3 sin⁻¹(x³)
2) ∫ x/√(x⁴ + a⁴). 1/2 log |x² + √(x⁴+ a⁴)|
3) ∫ x/√(4 - x⁴). 1/2 sin⁻¹(x²/2)
4) ∫ 2x/√(1- x²- x⁴). sin⁻¹{(2x²+1)/√5}
5) ∫ √{x/(a³ - x³)}. 2/3 sin⁻¹(x³⁾²/a³⁾²)
6) ∫ dx/{x²⁾³ √(x²⁾³ - 4)}. 3 log|x¹⁾³ + √(x²⁾³ - 4)|
7) ∫ sec²x/(√(16 + tan²x). log|tan x + √(16+ tan²x)|
8) ∫ sec²x/(√(4 + tan²x). log|tan x + √(4+ tan²x)|
9) ∫ cosx/√(sin²x - 2 sinx -3). log|(sinx -1) + √(sin²x - 2 sinx -3)|
10) ∫ cosx/(√(4 + sin²x). log|sin x + √(4+ sin²x)|
11) ∫ sinx/(√(4cos²x -1). - 1/2 log|2 cos x + √(4cos²x - 1)|
12) ∫ sin 8x/√{8+ sin⁴4x). 1/4 log|sin²4x + √(9+ sin⁴4x)|
13) ∫ cos 2x/√(sin²2x +8). 1/2 log|sin2x + √(sin²2x +8)|
14) ∫ sin 2x/√(sin⁴x + 4 sin²x -2).
log|sin²x + 2+ √(sin⁴x + 4 sin²x - 2)|
15) ∫ sin 2x/√(cos⁴x- sin²x+2). - log|(cos²x + 1/2) + √(cos⁴x + cos²x +1)|
16) ∫ cosx/√{4 - sin²x). sin⁻¹ {(sinx)/2}
17) ∫ cosx/√(sin²x- 2sinx -3). log|(sinx - 1) + 1 √(sin²x - 2 sinx -3)|
18) ∫ √(secx - 1) dx. - log|(cosx + 1/2) + √(cos²x+ cosx)|
19) ∫ √(cosecx - 1) dx. log|(sinx + 1/2) + √(sin²x + sin x)|.
20) ∫ √{sin(x- a)/sin(x+ a)}. - cos a sin⁻¹(cosx/cos a) - sin a. log|sinx + √(sin²x - sin²a)|
21) ∫ dx/{√(1- x²)(9+ (sin⁻¹x)²)}. Log|sin⁻¹x + √(9+ (sin⁻¹x)²)|
22) ∫ dx/{x √(log x)² -5}. log|log x +√(log x)² - 5|
23) ∫ dx/{x √(4 - (9log x)²}. 1/3 sin⁻¹{(3 log x)/2}
24) ∫ eˣ/√(4 - e²ˣ). sin⁻¹(eˣ/2)
25) ∫ eˣ/√(16 - e²ˣ). sin⁻¹(eˣ/4)
26) ∫ eˣ/√(5 - 4eˣ - e²ˣ). sin⁻¹ {(eˣ+2)/3}
27) ∫ aˣ/√(1 - a²ˣ). 1/(log a) sin⁻¹(aˣ)
INTEGRAL OF THE FORM :-
∫(px +q)/(ax²+ bx+ c)
Step 1: Write the numerator px + q in the following form:
px+ q = K{d/dx (ax²+ bx+c}+ M
i.e. px + q= K(2ax + b) + M
Step 2: Obtain the values of K and M by equating the coefficient of like powers of x on both sides.
Step 3: Replace px+q by K(2ax+b)+ M in the given integral to get
∫ (px +q)/(ax²+ bx+ c)
= K ∫(2ax +b)/(ax²+ bx+ c) + M ∫ dx/(ax²+ bx+ c).
Step 4: Integrate RHS in step 3 and put the values of K and M obtained in step 2.
EXERCISE-6
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1) ∫ x/(x²+ x+ 1). 1/2 log|x²+x+1| - 1/√3 tan⁻¹{(2x+1)/√3}
2) ∫ (4x+1)/(x²+ 3x+ 2). 2 log|x²+3x+2| - 5 log|{(x+1)/(x+2)}
3) ∫ (2x- 3)/(x²+ 3x -18). log|x²+3x -18| - 2/3 log|{(x- 3)/(x+6)}
4) ∫ x/(x²+ 3x+ 2). 1/2 log|x²+3x+2| - 3/2 log|{(x+1)/(x+2)}
5) ∫ (x+1)/(x²+ x+ 3). 1/2 log|x²+x+3| + 1/√11 tan⁻¹{( {(2x+1)/√11}
6) ∫ (x -3)/(x²+ 2x-4). 1/2 log|x²+2x-4| - 2/√5 log|{(x+1-√5)/(x+1+ √5)}
7) ∫ (2x -3)/(x²+ 6x +13). log|x²+6x+ 13| - 9/2 tan⁻¹{(x+3)/2}
8) ∫ (x -1)/(3x² -4x +3). 1/6 log|3x²- 4x +3| - √5/15 tan⁻¹{(3x-2)/√5}
9) ∫ 2x/(2+x -x²). - log|2+ x-x²| + 1/3 log|{(1+x)/(2-x)}
10) ∫ (1 - 3x)/(3x² +4x +2). - 1/2 log|3x²+ 4x +2| + 3/√2 tan⁻¹{(3x+2)/√2}
11) ∫ (2x +5)/(x² -x -2). log|x²- x-2| + 2 log|(x-2)/(x+1)|
12) ∫ (x³ +x)/(x⁴ - 9). 1/4 log|x⁴ - 9| + 1/12 log|{(x²- 3)/(x²+3)}
13) ∫ (ax³ +bx)/(x⁴ + c²). a/4 log|x⁴ + c²| + b/2c tan⁻¹{x²/c}
14) ∫ (x +2)/(2x² +6x +5). 1/4 log|2x²+ 6 x +5| + 1/2 tan⁻¹(2x+3)
15) ∫ (2 sin2x - cosx)/(6 - cos²x - 4 sinx). 2 log|sin²x - 4 sinx +5| + 7 tan⁻¹(sinx - 2)
16) ∫ (3 sin2x - 2) cosx)/(5 - cos²x - 4 sinx). 3 log|2 - sinx|+ 4/(2- sinx)
17) ∫ dx/(2eˣ + 3eˣ+1). -1/2 log|e⁻ˣ + 3e⁻ˣ+2)| + 3/2 log|(e⁻ˣ +1)/(e⁻ˣ+2)|
** INTEGRAL OF THE FORM ::
∫ P(x)/(ax²+ bx + c) where P(x) is a polynomial of degree greater than or equal to 2.
To evaluate this type of integral we divided the numerator by the denominator and express the integrand as
Q(x) + R(x)/(ax²+ bx + c). Where R(x) is a linear function of x.
∫ P(x)/(ax²+ bx + c) = ∫ Q(x) + ∫ R(x)/(ax²+ bx + c
Now to evaluate the second integral on RHS apply the method as Exerciss -6.
EXERCISE --7
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1) ∫ (x³ + x+1)/(x² -1). x²/2 + log|x² - 1| + 1/2 log|(x-1)/(x+1)|
2) ∫ (x² + x+1)/(x² -x). x + log|x² - x| + 2 log|(x-1)/x|
3) ∫ (x² + 5x+3)/(x²+3x +2). x + log|x² +3x+2| - 2 log|(x+1)/(x+2)|
4) ∫ (x² + x-1)/(x²+x -6). x + log|(x -2)/(x+3)|
5) ∫ (x² + 1)/(x²- 5x +6). x - 5 log|x - 2| + 10 log|(x - 3)|
6) ∫ (1- x²)/{x(1 - 2x)}. x/2 + log|x| - 3/4 log|(1- 2x)|
7) ∫ x²/(x²+7x +10). x - 7/2 log|x² +7x+ 10| + 29/6 log|(x+2)/(x+5)|
8) ∫ (x² + x+1)/(x²-x +1). x + log|x² - x+1| + 2/√3 tan⁻¹{(2x -1)/√3}
9) ∫ (x -1)²/(x²+ 2x +2). x - 2 log|x² + 2x+2| + 3 tan⁻¹{(x +1)}
10) ∫ (x³+x² + 2x+1)/(x²-x +1). x²/2 + 2x+ 3/2 log|x² - x+1| + 1/√3 tan⁻¹{(2x -1)/√3}
11) ∫ {x²(x⁴ +4)}/(x² +4). x⁵/5 - 4x³/3 + 20x - 40 tan⁻¹{(x/2}
12) ∫ x²/(x²+ 6x +12). x - 3 log|x² + 6x+12| + 2√3 tan⁻¹{(x+3)/√3}
*** INTEGRAL OF THE FORM
∫ (px +q)/√(ax²+ bx + c)
For Solution:
Step -1: Write the numerator px + q in the following form:
px +q = K{d/dx (ax² +bx +c)} + M
i. e., px + q= K(2ax+ b)+ M
Step-2: Obtain the values of K and M by equating the coefficient of like powers of x on both sides.
Step- 3: Replace px + q by K(2ax + b) + M in the given integral to get
∫ (px+q)/(ax²+ bx + c) =
K ∫ (2ax+b)/√(ax²+ bx + c) + M∫ dx/√(ax²+ bx + c)
Step-4: Integrate RHS in step 3 and put the values of K and M obtained in step 2
EXERCISE--8
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1) ∫ (2x+ 3)/√(x²+ 4x + 1). 2√(x²+ 4x +1) - log|x + 2 + √(x²+ 4x +1)|
2) ∫ (2x+ 1)/√(x²+ 2x -1). 2√(x²+ 2x -1) - log|x + 1+ √(x²+ 2x -1)|
3) ∫ (6x -5)/√(3x²- 5x + 1). 2√(3x²-5x +1)|
4) ∫ (x+ 2)/√(x²+ 2x - 1). √(x²+ 2x -1) + log|(x + 1) + √(x²+ 2x -1)|
5) ∫ (2x+ 5)/√(x²+ 2x + 5). 2√(x²+ 2x +5) + 3 log|x + 1 + √(x²+ 2x +5)|
6) ∫ (2x+ 1)/√(x²+ 4x + 3). 2√(x²+ 4x +3) - 3log|x + 2 + √(x²+ 4x +3)|
7) ∫ (2x+ 3)/√(x²+ 4x + 5). 2√(x²+ 4x +5) - log|x + 2 + √(x²+ 4x +5)|
8) ∫ (x+ 2)/√(x²+ 5x + 6). √(x²+ 5x +6) - 1/2 log|(x + 5/2) + √(x²+ 5x +6)|
9) ∫ x/√(x²+ x + 1). √(x²+ x +1) - 1/2 log|((2x + 1)/2 + √(x²+ x +1)|
10) ∫ (x+ 1)/√(4+ 5x- x²). - 4√(4 + 5x -x²) + 7/2 sin⁻¹{(2x-5)/√41}
11) ∫ (3x+ 1)/√(5 - 2x - x²). - 3√(5 - 2x -x²) - 2 sin⁻¹{(x +1)/√6}
12) ∫ x/√(8+ x - x²). - 8 √(8 +x -x²) + 1/2 sin⁻¹{(2x-1)/√33}
13) ∫ √{(1+x)/x}. √(x²+ x) + 1/2 log|(x+ 1/2) + √(x²+ x)|
14) ∫ √{(1- x)/(1+x)}. sin⁻¹x + √(1 - x²)
15) ∫ √{(a- x)/(a+x)}. a sin⁻¹{x/a} + √(a² - x²)
16) ∫ x. √{(a²- x²)/(a²+x²)}. a²/2 sin⁻¹{x²/a²} + 1/2 √(a⁴ - x⁴)
17) ∫ (x+2)/√(x² - 1)}. √(x²-1) + 2 log|x + √(x² -1)|
18) ∫ (x +1)/√(x² +1)}. √(x²+1) + log|x + √(x² +1)|
19) ∫ (x-1)/√(x² + 1)}. √(x²+1) - log|x + √(x² +1)|
20) ∫√{(1+x)/x}. √(x²+x) 1/2 log|(x+ 1/2)+ √(x²+x)|
*** INTEGRAL OF THE FORM
∫ dx/(a sin²x + b cos²x), OR
∫ dx/(a + bsin²x), OR
∫ dx/(a + b cos²x), OR
∫ dx/(a sinx + b cosx)² OR
∫ dx/(a + b sin²x + c cos²x)
For Solution::::
Step-1: Divide numerator and denominator both by cos²x.
Step-2: Replace sec²x, if any, in denominator by 1 + tan²x.
Step-3: Put tan x= t
so that sec²x dx = dt.
This substitution reduces the integral in the form of
∫ dt/(at²+ bt + c)
Step-4: Evaluate the integral obtained in step 3 by using the method discussed earlier.
EXERCISE--9
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1) ∫ dx/(a² sin²x+ b² cos²x. 1/ab
tan⁻¹{(a tanx)/b}
2) ∫ dx/(1+ 3sin²x+ 8cos²x. 1/6 tan⁻¹{(2 tanx)/3}
3) ∫ dx/(9sin²x+ 4cos²x. 1/6
tan⁻¹{(3tanx)/2}
4) ∫ dx/(4sin²x+ 5 cos²x. 1/2√5
tan⁻¹{(2 tanx)/√5}
5) ∫ dx/(2sinx+ 3cosx)². -1/2{(2tanx +3)}
6) ∫ dx/(3+ sin 2x). 1/2√2 tan⁻¹{(3 tanx +1)/2√2}
7) ∫ 2 dx/(2+ sin 2x). 2/√3 tan⁻¹{(2 tanx +1)/√3}
8) ∫ dx/(2 - 3 cos 2x). 1/2√5 log|{(√5 tanx -1)/(√5 tan x +1)}|
9) ∫ dx/(1+ 3 sin²x). 1/2 tan⁻¹{ (2tanx)}
10) ∫ dx/(3+ 2 cos²x). 1/√15 tan⁻¹{(√3 tanx)/√5}
11) ∫ dx/{(sin x- 2 cosx)(2 sinx+ cosx) 1/5 log|tanx -2))/(2 tanx +1)}|
12) ∫ sin 2x/(sin⁴x + cos⁴x). tan⁻¹{ tan²x}
13) ∫ dx/{cosx(sin x + 2 cosx). log|tan x +2|
14) ∫ dx/(sin²x + sin2x). 1/2 log|tanx/(tanx +2)|
15) ∫ dx/(cos2x + 3 sin²x). 1/√2 tan⁻¹{(√2 tanx}
16) ∫ sinx/sin3x. 1/2√3 log|(√3+ tanx)/(√3 - tanx)|
17) ∫ cosx/cos3x. 1/2√3 log|(1+ √3 tanx)/(1 - √3 tanx)|
**INTEGRAL OF THE FORM
∫ dx/(a sinx + b cos x),. OR
∫ dx/(a + b sin x),. OR
∫ dx/(a + b cos x),. OR
∫ dx/(a sinx + b cos x + c),.
For Solution:
Step-1: Put sinx = 2tan x/2/(1+ tan² x/2),
cosx= (1- tan² x/2)/(1+ tan² x/2)
Step-2: Replace 1+ tan²x/2 in the numerator by sec² x/2.
Step-3 : Put tan x/2 = t so that
1/2 sec² x/2 dx = dt.
This substitution reduces the integral in the form
∫ dt/(at² + bt +c),.
Step-4 : Evaluate the integral in STEP-3 by using method discussed earlier.
EXERCISE- 10
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1) ∫ dx/(2+ cos x). 2/√3 tan⁻¹{( tan x/2)/√3}
2) ∫ dx/(1 - 2sin x). 1/√3 log|(tan x/2 - 2 - √3)/(tan x/2 - 2 + √3)|
3) ∫ dx/(5+ 4cos x). 2/3 tan⁻¹{( tan x/2)/3}
4) ∫ dx/(5 - 4 sin x). 2/3 tan⁻¹{(5tan x/2 - 4)/3}
5) ∫ dx/(1- 2 sinx). 1/√3 log|( tan x/2 - 2 - √3)/(tan x/2 - 2 +√3)}|
6) ∫ dx/(4cos x -1). 1/√15 log|{(√3+ √5 tan x/2)/(√3 - √5 tan x/2)}|
7) ∫ dx/(1 + sinx + cos x). log|{( tan x/2) +1}
8) ∫ dx/(1 - sinx + cos x). - log|{(1- tan x/2)}|
9) ∫ dx/(3 + 2sinx + cos x). tan⁻¹{1+ tan x/2}
10) ∫ dx/(13 + 3sinx + 4cos x). 1/6 tan⁻¹{(5tan x/2 +2)}/6
11) ∫ dx/(cosx - sinx). 1/√2 log|(√2+ tan x/2 +1)/(√2 - tan x/2 -1)}|
12) ∫ dx/(sinx + cos x). 1/√2 log{(√2+ tan x/2 - 1)/(√2 - tan x/2 +1)}
13) ∫ dx/(2 + sinx + cos x). √2 tan⁻¹{1+ tan x/2}/√2
14) ∫ dx/(sinx + √3 cos x). 1/2 log|(1+ √3 tan x/2)/(3- √3 tan x/2)|
15) ∫ dx/(√3sinx + cos x). 1/2 log|{tan (x/2 + π/12)}
16) ∫ dx/(sinx - √3 cos x). 1/2 log|{tan (x/2 - π/6)}
17) ∫ dx/(5+ sinx + 7cos x). 1/5 log|(tan (x/2 +2)/(tan x/2 -3)}
18) ∫ (1+ sinx)/{sinx(1 + cos x)}. 1/2{ log|{tan x/2| + (tan² x/2)/2 + 2 tan x/2}.
ALTERNATIVE METHOD TO EVALUATE INTEGRAL OF THE FORM:
) ∫ dx/(asinx + bcos x).
For Solution, We substitute
a= r cos k, b= r sin k and so
r² = √(a² + b²), k= tan⁻¹(b/a)
So, a sink + b cos k
= r cos k sin k + r sin k cos k
= r sin(x+ k)
So, ∫ dx/(asinx + bcos x).
= 1/r ∫ dx/sin(x+k).
= 1/r ∫ cosec(x+k) dx.
= 1/r log|tan(x/2 + k/2)|+ C
= 1/√(a²+ b²) log|tan(x/2 + 1/2 tan⁻¹(b/a)|+ C.
EXERCISE-11
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1) ∫ dx/(√3 sinx + cos x). 1/2 log|tan(x/2 + π/12)|
2) ∫ dx/(sinx + √3 cos x). 1/2 log|tan(x/2 + π/6)|
*** INTEGRALS OF THE FORM:
∫ (a sinx + b cosx)/(c sinx + d cos x).
For Solution:
Step-1: Write
Numerator= K(Diff. of denominator) + M(Denominator).
i.e., a sin x + b cos x= K(c cos x - d sinx)+ M(c sinx + d cosx).
Step-2: Obtain the values of K and M by equating the coefficient of sin x and cosx on both the sides.
Step-3: Replace numerator in the integrand by K(c cosx - d sinx) + M(c sinx+ d cosx) to obtain
∫ (a sinx + b cosx)/(c sinx + d cos x). = K∫ (c cosx - d sinx)/(c sinx + d cos x) dx + M∫ (c sinx + d cosx)/(c sinx + d cos x).
= K log|c sinx + d cosx| + Mx + C
EXERCISE--12
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1) ∫ (3 sinx + 2 cosx)/(3 cosx + 2 sin x). -5x/13 + 12/13 log|3 cosx + 2 sinx |
2) ∫ dx/(1+ tanx). x/2 + 1/2 log|sinx + cosx|
3) ∫ dx/(1+ cotx). -1/2 log|sinx + cosx| + x/2
*** INTEGRALS OF THE FORM:
∫ (a sinx +b cosx + c)/(p sinx + q cos x + r).
For Solution:
Step-1: Write
Numerator= K(Diff. of denominator) + M(Denominator) + v.
i.e., a sin x + b cos x + c= K(p cos x - q sinx)+ M(p sinx + q cosx +r)+ v.
Step-2: Obtain the values of K and M by equating the coefficient of sin x and cosx and constant on both the sides.
Step-3: Replace numerator in the integrand by K(p cosx - q sinx) + M(p sinx+ q cosx + r)+ v to obtain
∫ (a sinx + b cosx+ c)/(p sinx + q cos x +r).
= K∫ (p cosx - q sinx)/(p sinx + q cos x + r) dx + M ∫ (p sinx + q cosx +r)/(p sinx + q cos x +r)+ v + ∫ dx/(p sinx + q cosx +r).
= K log|p sinx + q cosx + r| + Mx + v ∫ dx/(p sinx + q cosx +r).
Step-4: Evaluate the integral on RHS in step 3 by using the method discussed earlier.
EXERCISE- 13
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1) ∫ dx/(1- cotx). x/2 + 1/2 log|sinx - cosx|
2) ∫ dx/(1- tanx). x/2 - 1/2 log|sinx - cosx|
3) ∫ dx/(p + q tanx). px/(p²+ q²) + q/(p²+ q²) log|p cosx + q sinx|
4) ∫ dx/(4 cotx + 3). 3x/25 - 4/25 log|3sinx + 4 cosx|
5) ∫(2 tanx + 3)/(3 tanx+ 4). 18x/25 + 1/25 log|3 sinx + 4 cosx|
6) ∫ dx/(4 + 3 tan x). 4x/25 + 3/25 log|3sinx + 4 cosx|
7) ∫(8 cotx + 1)/(3cot x + 2). 2x + log|2sinx + 3 cosx|
8) ∫(3 cosx + 2)/(sinx + 2 cos x + 3). 6x/5 + 3/5 log|sinx + 2 cosx +3| + -8/5 tan⁻¹{(tan x/2 +1)/2}
9) ∫(3 + 2 cosx + 4)/(2sinx + cos x + 3). 2x - 3 tan⁻¹{(tan x/2 +1)}
10) ∫(5cosx + 6)/(sinx + 2 cos x + 3). 2x + log|sinx + 2 cosx +3|
11) ∫(3 cosx + 2 sinx)/(3sinx + 4 cos x). 18x/25 + 1/25 log|3sinx + 4 cosx|
12) ∫(5cosx + 4 sinx)/(5sinx + 4 cos x). 40x/41 + 9/41 log|5sinx + 4 cosx|
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