Rewrite the expression $\frac{x+3}{x^2+4x+5}$ inside the integral in factored form
$\int\frac{x+3}{1+\left(x+2\right)^2}dx$
2
We can solve the integral $\int\frac{x+3}{1+\left(x+2\right)^2}dx$ 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
3
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
$du=dx$
Intermediate steps
4
Rewriting $x$ in terms of $u$
$x=u-2$
Intermediate steps
5
Substituting $u$, $dx$ and $x$ in the integral and simplify
$\int\frac{1+u}{1+u^2}du$
6
Expand the fraction $\frac{1+u}{1+u^2}$ into $2$ simpler fractions with common denominator $1+u^2$
Expand the integral $\int\left(\frac{1}{1+u^2}+\frac{u}{1+u^2}\right)du$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
$\int\frac{1}{1+u^2}du+\int\frac{u}{1+u^2}du$
8
Rewrite the fraction $\frac{u}{1+u^2}$ inside the integral as the product of two functions: $u\frac{1}{1+u^2}$
$\int\frac{1}{1+u^2}du+\int u\frac{1}{1+u^2}du$
9
We can solve the integral $\int u\frac{1}{1+u^2}du$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
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