Integrate Using u-Substitution integral of (x^2)/((1+x^3)^2) with respect to x
The problem is asking for the evaluation of a definite integral using the method of u-substitution. Specifically, you are asked to find the integral of the function (x^2)/((1+x^3)^2) with respect to the variable x. The method of u-substitution involves substituting a part of the integrand (the expression inside the integral) with a new variable, usually denoted as 'u', which simplifies the integral and makes it easier to solve. The substitution is chosen such that the differential du is also present in the integrand, or can be easily obtained through algebraic manipulation, to replace the original variable 'x' and its differential 'dx'. After the substitution, you would integrate with respect to 'u' and then substitute back to the original variable 'x' to obtain the final result.
$\int \frac{x^{2}}{\left(\left(\right. 1 + x^{3} \left.\right)\right)^{2}} d x$
Answer
Solution:
Step:1
Choose $u = 1 + x^{3}$. Consequently, $d u = 3 x^{2} d x$, which implies $\frac{1}{3} d u = x^{2} d x$. Transition to $u$ and $d u$ for the integral.
Step:1.1
Set $u = 1 + x^{3}$. Compute $\frac{d u}{d x}$.
Step:1.1.1
Take the derivative of $1 + x^{3}$: $\frac{d}{d x} [1 + x^{3}]$.
Step:1.1.2
Utilize the Sum Rule: differentiate $1 + x^{3}$ as $\frac{d}{d x} [1] + \frac{d}{d x} [x^{3}]$.
Step:1.1.3
Since the derivative of a constant is zero: $0 + \frac{d}{d x} [x^{3}]$.
Step:1.1.4
Apply the Power Rule, which states $\frac{d}{d x} [x^{n}] = n x^{n - 1}$ for $n = 3$: $0 + 3 x^{2}$.
Step:1.1.5
Combine $0$ and $3 x^{2}$ to get $3 x^{2}$.
Step:1.2
Express the integral in terms of $u$ and $d u$: $\int \frac{1}{u^{2}} \cdot \frac{1}{3} d u$.
Step:2
Proceed to simplify.
Step:2.1
Combine $\frac{1}{u^{2}}$ with $\frac{1}{3}$: $\int \frac{1}{u^{2} \cdot 3} d u$.
Step:2.2
Reposition $3$ to precede $u^{2}$: $\int \frac{1}{3 u^{2}} d u$.
Step:3
Extract the constant $\frac{1}{3}$ from the integral: $\frac{1}{3} \int \frac{1}{u^{2}} d u$.
Step:4
Apply exponent rules.
Step:4.1
Rewrite $u^{2}$ in the numerator as a negative exponent: $\frac{1}{3} \int (u^{2})^{- 1} d u$.
Step:4.2
Perform exponent multiplication in $(u^{2})^{- 1}$.
Step:4.2.1
Invoke the power of a power rule, $(a^{m})^{n} = a^{m n}$: $\frac{1}{3} \int u^{2 \cdot - 1} d u$.
Step:4.2.2
Multiply $2$ by $- 1$: $\frac{1}{3} \int u^{- 2} d u$.
Step:5
Integrate $u^{- 2}$ using the Power Rule to obtain $- u^{- 1}$: $\frac{1}{3} (- u^{- 1} + C)$.
Step:6
Simplify further.
Step:6.1
Express $\frac{1}{3} (- u^{- 1} + C)$ as $\frac{1}{3} (- \frac{1}{u}) + C$.
Step:6.2
Distribute $\frac{1}{3}$ across $\frac{1}{u}$: $- \frac{1}{3 u} + C$.
Step:7
Substitute $u$ back with $1 + x^{3}$: $- \frac{1}{3 (1 + x^{3})} + C$.
Knowledge Notes:
U-Substitution: This is a technique used in integration, which involves changing the variable of integration to simplify the integral. It's akin to the reverse process of the chain rule in differentiation.
Sum Rule in Differentiation: This rule states that the derivative of a sum of two functions is the sum of their derivatives.
Power Rule in Differentiation: A fundamental rule that states the derivative of $x^n$ with respect to $x$ is $nx^{n-1}$.
Constant Multiple Rule: This rule allows us to take constants outside of an integral. If $k$ is a constant, then $\int k f(x) dx = k \int f(x) dx$.
Negative Exponents: A negative exponent means that the base is on the wrong side of the fraction line, so you flip the base to the other side. For example, $x^{-n} = \frac{1}{x^n}$.
Power Rule in Integration: This is used to integrate functions of the form $x^n$. The integral is $\frac{x^{n+1}}{n+1} + C$, except when $n = -1$, in which case the integral is $\ln|x| + C$.
Back-Substitution: After integrating with respect to $u$, we substitute back to the original variable to express the antiderivative in terms of the original variable.
