Problem to solve
Find the indefinite integral \[\int\sin[\ln[x]] dx\]
Solution
STEP 1
Take the integral: integral sin(log(x)) dx
STEP 2
For the integrand sin(log(x)), substitute u = log(x) and du = 1/x dx: = integral e^u sin(u) du
STEP 3
For the integrand e^u sin(u), use the formula integral exp(α u) sin(β u) du = (exp(α u) (-β cos(β u) + α sin(β u)))/(α^2 + β^2): = 1/2 e^u (sin(u) - cos(u)) + constant
STEP 4
Substitute back for u = log(x): = 1/2 x (sin(log(x)) - cos(log(x))) + constant
STEP 5
Which is equal to: Answer: | | = -1/2 x (cos(log(x)) - sin(log(x))) + constant
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Number
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Edited by tododemate® - 2021