Integrate Using u-Substitution integral of x^2sin(x^3) with respect to x

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 $\frac{d}{dx}{x}^n=n{x}^{n-1}$ 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:

1. 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.

2. 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.

3. Power Rule: This is a basic rule for differentiation which states that if $f(x)=x^n$, then $f^{\prime }(x)=n{x}^{n-1}$.

4. Integration of Trigonometric Functions: The integral of $\sin (x)$ with respect to $x$ is $-\cos (x)$, and this is a standard result in calculus.

5. Constants in Integration: Constants can be factored out of integrals since they do not affect the process of integration.

6. 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.

7. 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.