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 fraction $\frac{x^3}{\left(\sqrt{1+x}\right)^4}$ inside the integral as the product of two functions: $x^3\frac{1}{\left(\sqrt{1+x}\right)^4}$
We can solve the integral $\int x^3\frac{1}{\left(\sqrt{1+x}\right)^4}dx$ 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 $v$
We can solve the integral $\int\frac{1}{\left(1+x\right)^{2}}dx$ 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 $1+x$ 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
Substituting $u$ and $dx$ in the integral and simplify
Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Replace $u$ with the value that we assigned to it in the beginning: $1+x$
Now replace the values of $u$, $du$ and $v$ in the last formula
The integral $-3\int\frac{x^{2}}{-1-x}dx$ results in: $\frac{3}{2}x^2-3x+3\ln\left(-1-x\right)$
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 $C$
