We can solve the integral $\int\log \left(7x+10\right)dx$ 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 $7x+10$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
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 finding the derivative of the equation above
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{7}$ out of the integral
Apply the formula: $\int\log_{b}\left(x\right)dx$$=x\log_{b}\left(x\right)-\frac{x}{\ln\left(b\right)}+C$, where $b=10$ and $x=u$
Multiplying the fraction by $-1$
Replace $u$ with the value that we assigned to it in the beginning: $7x+10$
Simplify the product $-(7x+10)$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Expand and simplify
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