The integral cos(loge x )dx is equal to (where C is a constant of integration)

Engineering

Mathematics

Integration by Parts

Integration of Miscellaneous Functions

Integration by Substitution

Question

The integral $\int$ cos(log_e x )dx is equal to (where C is a constant of integration)

x[cos(log_e x) + sin(log_e x)] +C

$\frac{x}{2} [\cos (\log_{e} x) + \sin (\log_{e} x)] + C$

x[cos(log_e x) – sin(log_e x)] +C

$\frac{x}{2} [\sin (\log_{e} x) - \cos (\log_{e} x)] + C$

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I = $\int$ cos(lnx) dx = $\int$ 1.cos(lnx) dx

$∴ I = x . coslnx + \int x \cdot sinlnx \cdot \frac{1}{x} dx$

$∴$ I = x · cos (lnx) + $\int$ 1.sin(ln x) dx

$∴$ I = x · cos (lnx) + x · sin x (lnx) $- \int x \cdot coslnx \cdot \frac{1}{x} dx$

$∴$ 2I = x (cos lnx + sin lnx)

$∴ I = \frac{x}{2} (coslnx + sinlnx) + C$