Math / Calculus

Question $\int x^{2} \cos x^{3} d x=$

Studdy Solution

STEP 1

1. The integral $\int x^{2} \cos x^{3} \, dx$ is an indefinite integral.
2. We will use substitution to simplify the integral.

STEP 2

1. Identify a suitable substitution.
2. Perform the substitution.
3. Integrate the resulting expression.
4. Substitute back to the original variable.

STEP 3

Identify a suitable substitution. Notice that the derivative of $x^3$ is $3x^2$ , which is similar to the $x^2$ term in the integrand. Thus, let:
$u = x^3$
Differentiate both sides with respect to $x$ :
$\frac{du}{dx} = 3x^2$
Rearrange to express $dx$ in terms of $du$ :
$dx = \frac{du}{3x^2}$

STEP 4

Substitute $u = x^3$ and $dx = \frac{du}{3x^2}$ into the integral:
$\int x^{2} \cos(x^{3}) \, dx = \int x^{2} \cos(u) \frac{du}{3x^2}$
The $x^2$ terms cancel out:
$= \int \cos(u) \frac{1}{3} \, du$
$= \frac{1}{3} \int \cos(u) \, du$

STEP 5

Integrate the resulting expression:
$\frac{1}{3} \int \cos(u) \, du = \frac{1}{3} \sin(u) + C$
where $C$ is the constant of integration.

STEP 6

Substitute back to the original variable $x$ :
$\frac{1}{3} \sin(u) + C = \frac{1}{3} \sin(x^3) + C$
The solution to the integral is:
$\boxed{\frac{1}{3} \sin(x^3) + C}$

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