Math / Calculus

Question $\int \frac{3 x^{2}}{x^{3}+8} d x$

Studdy Solution

STEP 1

1. The integral $\int \frac{3 x^{2}}{x^{3}+8} \, dx$ involves a rational function.
2. The denominator $x^3 + 8$ can be factored or simplified.
3. The numerator $3x^2$ suggests a potential substitution method for integration.

STEP 2

1. Factor or simplify the denominator $x^3 + 8$ .
2. Use an appropriate substitution to simplify the integral.
3. Integrate the simplified function.
4. Substitute back to return to the original variable.

STEP 3

Recognize that the denominator $x^3 + 8$ can be factored using the sum of cubes formula.
$x^3 + 8 = (x + 2)(x^2 - 2x + 4)$

STEP 4

Notice that the numerator $3x^2$ is the derivative of the denominator $x^3 + 8$ . Use the substitution $u = x^3 + 8$ .
Let: $u = x^3 + 8$ Then: $du = 3x^2 \, dx$

STEP 5

Rewrite the integral in terms of the new variable $u$ .
$\int \frac{3 x^{2}}{x^{3}+8} \, dx = \int \frac{1}{u} \, du$

STEP 6

Integrate the simplified function $\int \frac{1}{u} \, du$ .
$\int \frac{1}{u} \, du = \ln |u| + C$

STEP 7

Substitute back $u = x^3 + 8$ to return to the original variable.
$\ln |u| + C = \ln |x^3 + 8| + C$
Solution: $\int \frac{3 x^{2}}{x^{3}+8} \, dx = \ln |x^3 + 8| + C$

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Question $\int \frac{3 x^{2}}{x^{3}+8} d x$

Studdy Solution

STEP 1

1. The integral $\int \frac{3 x^{2}}{x^{3}+8} \, dx$ involves a rational function.
2. The denominator $x^3 + 8$ can be factored or simplified.
3. The numerator $3x^2$ suggests a potential substitution method for integration.

STEP 2

1. Factor or simplify the denominator $x^3 + 8$ .
2. Use an appropriate substitution to simplify the integral.
3. Integrate the simplified function.
4. Substitute back to return to the original variable.

STEP 3

Recognize that the denominator $x^3 + 8$ can be factored using the sum of cubes formula.
$x^3 + 8 = (x + 2)(x^2 - 2x + 4)$

STEP 4

Notice that the numerator $3x^2$ is the derivative of the denominator $x^3 + 8$ . Use the substitution $u = x^3 + 8$ .
Let: $u = x^3 + 8$ Then: $du = 3x^2 \, dx$

STEP 5

Rewrite the integral in terms of the new variable $u$ .
$\int \frac{3 x^{2}}{x^{3}+8} \, dx = \int \frac{1}{u} \, du$

STEP 6

Integrate the simplified function $\int \frac{1}{u} \, du$ .
$\int \frac{1}{u} \, du = \ln |u| + C$

STEP 7

Substitute back $u = x^3 + 8$ to return to the original variable.
$\ln |u| + C = \ln |x^3 + 8| + C$
Solution: $\int \frac{3 x^{2}}{x^{3}+8} \, dx = \ln |x^3 + 8| + C$

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Question∫3x2x3+8dx\int \frac{3 x^{2}}{x^{3}+8} d x∫x3+83x2​dx

Studdy Solution

STEP 1

1. The integral ∫3x2x3+8 dx\int \frac{3 x^{2}}{x^{3}+8} \, dx∫x3+83x2​dx involves a rational function.2. The denominator x3+8x^3 + 8x3+8 can be factored or simplified.3. The numerator 3x23x^23x2 suggests a potential substitution method for integration.

STEP 2

1. Factor or simplify the denominator x3+8x^3 + 8x3+8.2. Use an appropriate substitution to simplify the integral.3. Integrate the simplified function.4. Substitute back to return to the original variable.

STEP 3

Recognize that the denominator x3+8x^3 + 8x3+8 can be factored using the sum of cubes formula.x3+8=(x+2)(x2−2x+4) x^3 + 8 = (x + 2)(x^2 - 2x + 4) x3+8=(x+2)(x2−2x+4)

STEP 4

Notice that the numerator 3x23x^23x2 is the derivative of the denominator x3+8x^3 + 8x3+8. Use the substitution u=x3+8u = x^3 + 8u=x3+8.Let: u=x3+8 u = x^3 + 8 u=x3+8 Then: du=3x2 dx du = 3x^2 \, dx du=3x2dx

STEP 5

Rewrite the integral in terms of the new variable uuu.∫3x2x3+8 dx=∫1u du \int \frac{3 x^{2}}{x^{3}+8} \, dx = \int \frac{1}{u} \, du ∫x3+83x2​dx=∫u1​du

STEP 6

Integrate the simplified function ∫1u du\int \frac{1}{u} \, du∫u1​du.∫1u du=ln⁡∣u∣+C \int \frac{1}{u} \, du = \ln |u| + C ∫u1​du=ln∣u∣+C

STEP 7

Substitute back u=x3+8u = x^3 + 8u=x3+8 to return to the original variable.ln⁡∣u∣+C=ln⁡∣x3+8∣+C \ln |u| + C = \ln |x^3 + 8| + C ln∣u∣+C=ln∣x3+8∣+CSolution: ∫3x2x3+8 dx=ln⁡∣x3+8∣+C \int \frac{3 x^{2}}{x^{3}+8} \, dx = \ln |x^3 + 8| + C ∫x3+83x2​dx=ln∣x3+8∣+C

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Question $\int \frac{3 x^{2}}{x^{3}+8} d x$

1. The integral $\int \frac{3 x^{2}}{x^{3}+8} \, dx$ involves a rational function.
2. The denominator $x^3 + 8$ can be factored or simplified.
3. The numerator $3x^2$ suggests a potential substitution method for integration.

1. Factor or simplify the denominator $x^3 + 8$ .
2. Use an appropriate substitution to simplify the integral.
3. Integrate the simplified function.
4. Substitute back to return to the original variable.

Recognize that the denominator $x^3 + 8$ can be factored using the sum of cubes formula.
$x^3 + 8 = (x + 2)(x^2 - 2x + 4)$

Notice that the numerator $3x^2$ is the derivative of the denominator $x^3 + 8$ . Use the substitution $u = x^3 + 8$ .
Let: $u = x^3 + 8$ Then: $du = 3x^2 \, dx$

Rewrite the integral in terms of the new variable $u$ .
$\int \frac{3 x^{2}}{x^{3}+8} \, dx = \int \frac{1}{u} \, du$

Integrate the simplified function $\int \frac{1}{u} \, du$ .
$\int \frac{1}{u} \, du = \ln |u| + C$

Substitute back $u = x^3 + 8$ to return to the original variable.
$\ln |u| + C = \ln |x^3 + 8| + C$
Solution: $\int \frac{3 x^{2}}{x^{3}+8} \, dx = \ln |x^3 + 8| + C$