Math / Calculus

QuestionIf we perform the appropriate (i.e. helpful) $u$ -sub for $\int x^{2} \sin \left(x^{3}\right) \mathrm{dx}$ , what does the new integral look like in terms of $u$ right after performing the substitution? $\int x^{2} \sin \left(u^{3}\right) d u$ $\int \frac{1}{3} \sin (u) d u$ none of these $\int u^{2} \sin \left(u^{3}\right) d u$

Studdy Solution

STEP 1

1. We are given the integral $\int x^{2} \sin \left(x^{3}\right) \mathrm{dx}$ .
2. We need to perform a $u$ -substitution to simplify the integral.
3. The goal is to express the integral in terms of $u$ .

STEP 2

1. Identify the appropriate substitution for $u$ .
2. Differentiate the substitution to find $du$ .
3. Rewrite the integral in terms of $u$ and $du$ .

STEP 3

Identify the appropriate substitution. Since the integrand contains $\sin(x^3)$ , a helpful substitution is:
$u = x^3$

STEP 4

Differentiate the substitution to find $du$ .
$\frac{du}{dx} = 3x^2$
Thus,
$du = 3x^2 \, dx$

STEP 5

Solve for $dx$ in terms of $du$ and $x$ :
$dx = \frac{du}{3x^2}$

STEP 6

Substitute $u = x^3$ and $dx = \frac{du}{3x^2}$ into the original integral:
$\int x^2 \sin(x^3) \, dx = \int x^2 \sin(u) \cdot \frac{du}{3x^2}$
The $x^2$ terms cancel out, simplifying to:
$\int \frac{1}{3} \sin(u) \, du$
The new integral in terms of $u$ is:
$\int \frac{1}{3} \sin(u) \, du$
The correct choice is:
$\int \frac{1}{3} \sin(u) \, du$

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Studdy Solution

STEP 1

1. We are given the integral $\int x^{2} \sin \left(x^{3}\right) \mathrm{dx}$ .
2. We need to perform a $u$ -substitution to simplify the integral.
3. The goal is to express the integral in terms of $u$ .

STEP 2

1. Identify the appropriate substitution for $u$ .
2. Differentiate the substitution to find $du$ .
3. Rewrite the integral in terms of $u$ and $du$ .

STEP 3

Identify the appropriate substitution. Since the integrand contains $\sin(x^3)$ , a helpful substitution is:
$u = x^3$

STEP 4

Differentiate the substitution to find $du$ .
$\frac{du}{dx} = 3x^2$
Thus,
$du = 3x^2 \, dx$

STEP 5

Solve for $dx$ in terms of $du$ and $x$ :
$dx = \frac{du}{3x^2}$

STEP 6

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Studdy Solution

STEP 1

1. We are given the integral ∫x2sin⁡(x3)dx\int x^{2} \sin \left(x^{3}\right) \mathrm{dx}∫x2sin(x3)dx.2. We need to perform a uuu-substitution to simplify the integral.3. The goal is to express the integral in terms of uuu.

STEP 2

1. Identify the appropriate substitution for uuu.2. Differentiate the substitution to find dududu.3. Rewrite the integral in terms of uuu and dududu.

STEP 3

Identify the appropriate substitution. Since the integrand contains sin⁡(x3)\sin(x^3)sin(x3), a helpful substitution is:u=x3 u = x^3 u=x3

STEP 4

Differentiate the substitution to find dududu. dudx=3x2 \frac{du}{dx} = 3x^2 dxdu​=3x2Thus, du=3x2 dx du = 3x^2 \, dx du=3x2dx

STEP 5

Solve for dxdxdx in terms of dududu and xxx:dx=du3x2 dx = \frac{du}{3x^2} dx=3x2du​

STEP 6

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1. We are given the integral $\int x^{2} \sin \left(x^{3}\right) \mathrm{dx}$ .
2. We need to perform a $u$ -substitution to simplify the integral.
3. The goal is to express the integral in terms of $u$ .

1. Identify the appropriate substitution for $u$ .
2. Differentiate the substitution to find $du$ .
3. Rewrite the integral in terms of $u$ and $du$ .

Identify the appropriate substitution. Since the integrand contains $\sin(x^3)$ , a helpful substitution is:
$u = x^3$

Differentiate the substitution to find $du$ .
$\frac{du}{dx} = 3x^2$
Thus,
$du = 3x^2 \, dx$

Solve for $dx$ in terms of $du$ and $x$ :
$dx = \frac{du}{3x^2}$