Expand the integral $\int\left(3x^{3}+6x^{2}+7x+16+\frac{31}{x-2}\right)dx$ into $5$ integrals using the sum rule for integrals, to then solve each integral separately
We can solve the integral $\int\frac{31}{x-2}dx$ 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 $x-2$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
$u=x-2$
Intermediate steps
5
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
$du=dx$
6
Substituting $u$ and $dx$ in the integral and simplify
Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more