Question

ate\int log \: x \: dx" alt=" \sf Evaluate\int log \: x \: dx" align="absmiddle" class="latex-formula">
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Thenkewww ☃️


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Answer 1

Answer:

x(log(x) - 1) + C

Step-by-step explanation:

The formula for the derivative of a logarithm function is:

$\frac{d}{dx} log_{a}(x) = \frac{1}{xlog(a)}$

For your case, the base is 10. Now that we have the derivative, lets use integration by parts. I will be differentiating log(x) and integrating 1:

u = log(x)

du = dx/xlog(10)

dv = 1

v = x

You get:

log(x)x - integral(dx/log(10))

Therefore the answer is:

log(x)x - x/log(10) + C

x(log(x) - 1) + C

Answer 2

$\sf \huge \underline \pink{ \purple {Answer : }}$

$\: \:$

$\: \: \: \: \: \: \: \rm \large log \: xdx = \int \: log \: x *\: 1dx$

$\:$

$\: \: \: \: \: \: \: \rm \large = log \: x \int 1 \: dx - \int \: ( \int \: 1dx \: \frac{d}{dx} log \: x)dx$

$\: \:$

$\: \: \: \: \: \: \: \rm \large \: = x \: log \: x - x + c$

$\: \: \:$

$\: \: \: \: \: \: \: \rm \large \: = log \: x \: (x) - \int \: x \: \frac{1}{x} dx$

$\: \:$

$\: \: \: \: \: \: \: \rm \large = x \: log \: x - \int \: 1dx$

$\: \:$

$\: \: \: \: \: \: \ \rm \color{red} \bold{\large = x(log \: x - 1) + c}$

$\: \: \: \: \: \: \: \:$

Hope Helps!