Integrate Using u-Substitution integral of x^2sin(x^3) with respect to x
The problem asks to perform a definite or indefinite integration on the provided function, x^2sin(x^3), by applying the u-substitution technique. This technique involves substituting a part of the integrand with a new variable, typically denoted as 'u', to simplify the integral into a form that is easier to evaluate. The first step is to identify a function within the integrand whose derivative also appears elsewhere in the integral. This chosen function is then set to 'u', and its derivative (du) is used to replace the corresponding part of the differential dx. The integral is then re-expressed in terms of 'u' and solved. Once the integral is found in terms of 'u', the variable 'u' is substituted back with the original function it replaced, giving the solution in terms of the original variable x.
$\int x^{2} sin \left(\right. x^{3} \left.\right) d x$
Answer
Solution:
Step 1
Assign $u = x^3$. Consequently, $du = 3x^2 dx$, which implies $\frac{1}{3}du = x^2 dx$. Substitute $u$ and $du$ into the integral.
Step 1.1
Set $u = x^3$ and compute $\frac{du}{dx}$.
Step 1.1.1
Take the derivative of $x^3$: $\frac{d}{dx}x^3$.
Step 1.1.2
Apply the Power Rule for differentiation, which states $\frac{d}{dx}x^n = nx^{n-1}$ for $n = 3$, yielding $3x^2$.
Step 1.2
Express the integral in terms of $u$ and $du$: $\int sin(u) \frac{1}{3} du$.
Step 2
Merge the sine function with the constant $\frac{1}{3}$: $\int \frac{sin(u)}{3} du$.
Step 3
Extract the constant $\frac{1}{3}$ from the integral: $\frac{1}{3}\int sin(u) du$.
Step 4
Integrate $sin(u)$ with respect to $u$ to obtain $-cos(u)$: $\frac{1}{3}(-cos(u) + C)$.
Step 5
Proceed to simplify the expression.
Step 5.1
Simplify to obtain: $\frac{1}{3}(-cos(u)) + C$.
Step 5.2
Combine the constant $\frac{1}{3}$ with the cosine function: $-\frac{cos(u)}{3} + C$.
Step 6
Substitute back $u$ with $x^3$: $-\frac{cos(x^3)}{3} + C$.
Step 7
Rearrange the terms to finalize the result: $-\frac{1}{3}cos(x^3) + C$.
Knowledge Notes:
The process of solving an integral using u-substitution involves several steps and knowledge of different calculus concepts:
u-Substitution: This is a technique used to simplify integrals by substituting a part of the integrand with a new variable $u$. This is particularly useful when the integrand is a composite function.
Derivative: The derivative of a function gives the rate at which the function value changes with respect to changes in the variable. It is a fundamental concept in calculus.
Power Rule: This is a basic rule for differentiation which states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Integration of Trigonometric Functions: The integral of $\sin(x)$ with respect to $x$ is $-\cos(x)$, and this is a standard result in calculus.
Constants in Integration: Constants can be factored out of integrals since they do not affect the process of integration.
Back-Substitution: After integrating with respect to $u$, it is necessary to substitute back the original variable to express the antiderivative in terms of the original variable.
Indefinite Integral and Constant of Integration: When computing an indefinite integral, a constant of integration ($C$) is added to represent the family of all antiderivatives.
By understanding and applying these concepts, one can solve integrals that may initially seem complex.
