Integrate the following and evaluate ∫2sinx/(3sinx+4cosx)
Mahima , 8 Years ago
Grade 12
Integral Calculus> Integrate the following and evaluate ∫2si...
1 Answers
Arun
You want to split it up in a very specific way. On the one hand, we can integrate the function:
(3cos(x) + 4sin(x)) / (3cos(x) + 4sin(x)) = 1
really easily (it just integrates to x). On the other hand, we can integrate the function:
(-3sin(x) + 4cos(x)) / (3cos(x) + 4sin(x))
easily as well, because a substitution of u = 3cos(x) + 4sin(x) yields an integral of ln|3cos(x) + 4sin(x)|. So, if we can express the numerator 2sin(x) + 3cos(x) as a linear combination of 3cos(x) + 4sin(x) and -3sin(x) + 4cos(x), then we can split the fraction up, and integrate each part easily.
Suppose A and B are such that:
A(3cos(x) + 4sin(x)) + B(-3sin(x) + 4cos(x)) = 2sin(x) + 3cos(x)
for all x. Then, equating coefficients:
4A - 3B = 2
3A + 4B = 3
Solving simultaneously yields:
A = 17/25
B = 6/25
Therefore the function is equal to:
17/25 + (6/25) * (-3sin(x) + 4cos(x)) / (3cos(x) + 4sin(x))
which integrates to:
(17x + 6ln|3cos(x) + 4sin(x)|) / 25 + C
(3cos(x) + 4sin(x)) / (3cos(x) + 4sin(x)) = 1
really easily (it just integrates to x). On the other hand, we can integrate the function:
(-3sin(x) + 4cos(x)) / (3cos(x) + 4sin(x))
easily as well, because a substitution of u = 3cos(x) + 4sin(x) yields an integral of ln|3cos(x) + 4sin(x)|. So, if we can express the numerator 2sin(x) + 3cos(x) as a linear combination of 3cos(x) + 4sin(x) and -3sin(x) + 4cos(x), then we can split the fraction up, and integrate each part easily.
Suppose A and B are such that:
A(3cos(x) + 4sin(x)) + B(-3sin(x) + 4cos(x)) = 2sin(x) + 3cos(x)
for all x. Then, equating coefficients:
4A - 3B = 2
3A + 4B = 3
Solving simultaneously yields:
A = 17/25
B = 6/25
Therefore the function is equal to:
17/25 + (6/25) * (-3sin(x) + 4cos(x)) / (3cos(x) + 4sin(x))
which integrates to:
(17x + 6ln|3cos(x) + 4sin(x)|) / 25 + C
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