What is the integral of $$x^2 \\sqrt[]{5+2x^3} \, dx$$ ?

To solve this integral, we can use substitution. Let $$u = 5 + 2x^3$$ .

What is the substitution for $$u$$ in the integral $$\int x^2 \\sqrt[]{5+2x^3} \, dx$$ ?

The correct substitution is $$u = 5 + 2x^3$$ .

What is the integral $$\int x^2 \\sqrt[]{5+2x^3} \, dx$$ in terms of $$u$$ ?

After substitution, the integral becomes $$\int \frac{1}{6} \sqrt{u} \, du$$ .

What is the result of $$\int \frac{1}{6} \sqrt{u} \, du$$ ?

The result is $$\frac{2}{9} u^{3/2} + c$$ .

What is the final answer for the integral $$\int x^2 \\sqrt[]{5+2x^3} \, dx$$ in terms of $$x$$ ?

The final answer is $$\frac{\sqrt{(5+2x^3)^3}}{9} + c$$ .

What does the constant $$c$$ represent in integration?

The constant $$c$$ represents the constant of integration, accounting for all possible antiderivatives.

What is the derivative of $$u = 5 + 2x^3$$ with respect to $$x$$ ?

The derivative is $$\frac{du}{dx} = 6x^2$$ .

How do you express $$dx$$ in terms of $$du$$ when using substitution?

From $$\frac{du}{dx} = 6x^2$$ , we have $$dx = \frac{du}{6x^2}$$ .

What is the integral of $$\sqrt{u}$$ ?

The integral is $$\frac{2}{3} u^{3/2} + c$$ .

What is the importance of the limits of integration in definite integrals?

Limits of integration define the interval over which the function is being integrated.

What is the relationship between differentiation and integration?

Differentiation and integration are inverse processes; integration reverses differentiation.

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus links differentiation and integration, stating that if $$F$$ is an antiderivative of $$f$$ , then $$\int_a^b f(x) \, dx = F(b) - F(a)$$ .

What is the geometric interpretation of an integral?

An integral represents the area under the curve of a function on a given interval.

What is the significance of the square root in the integral $$\int x^2 \\sqrt[]{5+2x^3} \, dx$$ ?

The square root indicates that the function being integrated has a non-linear component, affecting the shape of the area under the curve.

CuDSIA (3) Flashcards

Student preview

15 questions

Show all answers

1.

FLASHCARD QUESTION

Front

Back

2.

FLASHCARD QUESTION

Front

Back

3.

FLASHCARD QUESTION

Front

Back

4.

FLASHCARD QUESTION

Front

Back

5.

FLASHCARD QUESTION

Front

Back

6.

FLASHCARD QUESTION

Front

Back

7.

FLASHCARD QUESTION

Front

What is the purpose of substitution in integration?

Back

Substitution simplifies the integral by changing variables to make the integral easier to solve.

Create a free account and access millions of resources

Create resources

Host any resource

Get auto-graded reports

Continue with Google

Continue with Email

Continue with Classlink

Continue with Clever

or continue with

Microsoft

Apple

Others

Already have an account?

Similar Resources on Wayground

10 questions

Strategies to Avoid Communication Breakdown

Flashcard

•

11th - 12th Grade

10 questions

Marketing Mix - Place and Distribution

Flashcard

•

10th Grade - University

15 questions

What is the Limit?

Flashcard

•

11th Grade

12 questions

DNA Replication and Protein Synthesis

Flashcard

•

9th - 12th Grade

10 questions

Interpret Bar Graphs

Flashcard

•

12th Grade

9 questions

Лазерлер туралы жалпы түсінік

Flashcard

•

KG

6 questions

Unit 3.8 Market Structure

Flashcard

•

11th Grade

Popular Resources on Wayground

10 questions

Forest Self-Management

Lesson

•

1st - 5th Grade

25 questions

Multiplication Facts

Quiz

•

5th Grade

30 questions

Thanksgiving Trivia

Quiz

•

9th - 12th Grade

30 questions

Thanksgiving Trivia

Quiz

•

6th Grade

11 questions

Would You Rather - Thanksgiving

Lesson

•

KG - 12th Grade

48 questions

The Eagle Way

Quiz

•

6th Grade

10 questions

Identifying equations

Quiz

•

KG - University

10 questions

Thanksgiving

Lesson

•

5th - 7th Grade

Discover more resources for Mathematics

10 questions

Identifying equations

Quiz

•

KG - University