Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Integrate by parts
- Integrate by partial fractions
- Integrate by substitution
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- FOIL Method
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Rewrite the expression $\frac{x+3}{x^2+4x+5}$ inside the integral in factored form
We can solve the integral $\int\frac{x+3}{\left(x+2\right)^2+1}dx$ 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
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
Rewriting $x$ in terms of $u$
Substituting $u$, $dx$ and $x$ in the integral and simplify
Expand the fraction $\frac{u+1}{u^2+1}$ into $2$ simpler fractions with common denominator $u^2+1$
Expand the integral $\int\left(\frac{u}{u^2+1}+\frac{1}{u^2+1}\right)du$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
Rewrite the fraction $\frac{u}{u^2+1}$ inside the integral as the product of two functions: $u\frac{1}{u^2+1}$
We can solve the integral $\int u\frac{1}{u^2+1}du$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
Solve the integral by applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$
Now replace the values of $u$, $du$ and $v$ in the last formula
Replace $u$ with the value that we assigned to it in the beginning: $x+2$
The integral $-\int\arctan\left(u\right)du$ results in: $\left(-x-2\right)\arctan\left(x+2\right)+\frac{1}{2}\ln\left|1+\left(x+2\right)^2\right|$
Gather the results of all integrals
The integral $\int\frac{1}{u^2+1}du$ results in: $\arctan\left(x+2\right)$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$