Problem 1 :
Integrate x/√(x2+3)
Solution :
Let t = x2+3
differentiate with respect to x
dt = 2x dx
x dx = dt/2
= ∫ x/√(x2+3) dx
= ∫ (dt/2)√t
= (1/2) ∫√t dt
= (1/2) ∫t(1/2) dt
= (1/2) (t3/2 / (3/2) + C
= (1/2) (2/3) t3/2 + C
= (1/3) (x2+3)3/2 + C
Problem 2 :
Integrate (2 x + 3) √(x2+3x-5)
Solution :
Let t = x2+3x-5
differentiate with respect to x
dt = (2x+3) dx
= ∫(2x+3) √(x2+3x-5) dx
= ∫(dt/√t)
= ∫t(-1/2) dt
= t(1/2)/(1/2) + C
= 2√(x2+3x-5) + C
Problem 3 :
Integrate tan x
Solution :
= ∫tan x dx
= ∫(sin x/cos x) dx
t = cos x
differentiating with respect to x, we get
dt = -sin x dx
sin x dx = - dt
= ∫(-dt/t)
= -∫(1/t) dt
= -log t + C
= -log (cos x) + C
Problem 4 :
Integrate sec x
Solution :
= ∫sec x dx
= ∫ [sec x (sec x+tan x)/(sec x+tan x)] dx
= ∫[(sec2 x + sec x tan x)/(sec x + tan x)] dx
t = sec x + tan x
dt = sec x tan x + sec ² x
= ∫dt/t
= ∫(1/t) dt
= log t + C
= log (sec x + tan x) + C
Problem 5 :
Integrate x5 (1 + x6)7
Solution :
Let t = 1 + x⁶
differentiate with respect to x
dt = 6 x⁵ dx
dt/6 = x⁵ dx
x⁵ dx = dt/6
= ∫ x⁵ (1 + x⁶)⁷ dx
= ∫ t⁷ (dt/6)
= (1/6) t⁷ dt
= (1/6) [t(7+1)/(7+1)] + C
= (1/6) (t8/8) + C
= (1/48) t8 + C
= t8/48 + C
= (1 + x⁶)⁸/48 + C
Problem 6 :
Integrate (2Lx + m)/(Lx² + mx + n)
Solution :
Let t = (Lx2+mx+n)
differentiate with respect to x
dt = (2Lx + m) dx
= ∫ (dt/t)
= log t + C
= log (Lx² + mx + n) + C
Problem 7 :
Integrate (4ax + 2b)/(ax2 + bx + c)10
Solution :
t = ax2+bx+c
differentiate with respect to x
dt = 2ax+b
= ∫(4ax+2b)/(ax2+bx+c)10 dx
now we are going to take 2 from the numerator
= ∫ 2 (2ax+2b)/(ax2+bx+c)10 dx
= ∫2 (dt/t10)
= ∫2 t-10 dt
= 2t(-10+1)/(-10+1) + C
= 2t-9/(-9) + C
= (-2/9) (ax2 + bx + c)^(-9) + C
= [-2/9(ax2+bx+c)9] + C
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