Here, we show you a step-by-step solved example of integrals involving logarithmic functions. This solution was automatically generated by our smart calculator:
We can solve the integral $\int\ln\left(2x\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Differentiate both sides of the equation $u=2x$
Find the derivative
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Rearrange the equation
Divide both sides of the equation by $2$
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{2}$ out of the integral
The integral of the natural logarithm is given by the following formula, $\displaystyle\int\ln(x)dx=x\ln(x)-x$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
Multiply $-1$ times $2$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
Replace $u$ with the value that we assigned to it in the beginning: $2x$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Multiply the single term $\frac{1}{2}$ by each term of the polynomial $\left(2x\ln\left(2x\right)-2x\right)$
Simplifying
Divide $2$ by $2$
Divide $-2$ by $2$
Expand and simplify
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