(P1T4Q2) Integration By Parts [CLASS//QUIZIZZ]

(P1T4Q2) Integration By Parts [CLASS//QUIZIZZ]

Integrate

Math Response

$$\int\ln\left(x\right)\cdot x^3dx$$

Step 1.1: Choose "u"

(use LIATE)

u=___

Type answer here

Correct Answer: \ln\left(x\right)

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Math Response

$$\int\ln\left(x\right)\cdot x^3dx$$

Step 1.2: Choose "dv"

dv=___dx

Type answer here

Correct Answer: x^3

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Math Response

$$\int\ln\left(x\right)\cdot x^3dx$$

Step 2.1: Calculate "du"

du=___dx

Type answer here

Correct Answer: \frac{1}{x}

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Math Response

$$\int\ln\left(x\right)\cdot x^3dx$$

Step 2.2: Calculate "v"

v=___

Type answer here

Correct Answer: \frac{x^4}{4}

Deg^°

Rad

Reorder

Reorder the following:

What's the formula for Integration by Parts?

• $$u$$

• $$v$$

• $$-\int v$$

• $$du$$

1

2

3

4

• 1
• 2
• 3
• 4

Multiple Choice

$$uv-\int vdu$$

Step 3.1: Substitute in the formula

• $$\left(\ln\left(x\right)\right)\left(\frac{x^4}{4}\right)-\int\left(\frac{x^4}{4}\right)\left(\frac{1}{x}dx\right)$$

• $$\left(\ln\left(x\right)\right)\left(x^3\right)-\int\left(\frac{1}{x}\right)\left(\frac{x^2}{2}dx\right)$$

Multiple Choice

$$\left(\ln\left(x\right)\right)\left(\frac{x^4}{4}\right)-\int\left(\frac{x^4}{4}\right)\left(\frac{1}{x}dx\right)$$

Step 3.2: Simplification

• $$\frac{x^4\cdot\ln\left(x\right)}{4}-\frac{1}{4}\int x^4dx$$

• $$\frac{x^4-\ln\left(x\right)}{4}-4\int x^4dx$$

Multiple Choice

$$\frac{x^4\cdot\ln\left(x\right)}{4}-\frac{1}{4}\int x^4dx$$

Step 3.3: Solving the remaining integral

• $$\frac{x^4\cdot\ln\left(x\right)}{4}-\frac{1}{4}\left(\frac{x^5}{5}+C\right)$$

• $$\frac{x^4\cdot\ln\left(x\right)}{4}-\frac{1}{4}\left(4x^3+C\right)$$

Math Response

$$\frac{x^4\cdot\ln\left(x\right)}{4}-\frac{1}{4}\left(\frac{x^5}{5}+C\right)$$

Step 3.4: Write down the final answer

$$\int\ln\left(x\right)\cdot x^3dx=$$ _____+C

Type answer here

Correct Answer: \frac{x^2\times\ln\left(x\right)}{4}-\frac{1}{20}x^5

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Integrate

Math Response

$$\int8x\cdot\cos\left(2x\right)dx$$

Step 1.1: Choose "u"

(use LIATE)

u=___

Type answer here

Correct Answer: 8x

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Math Response

$$\int8x\cdot\cos\left(2x\right)dx$$

Step 1.2: Choose "dv"

dv=___dx

Type answer here

Correct Answer: \cos\left(2x\right)

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Math Response

$$\int8x\cdot\cos\left(2x\right)dx$$

Step 2.1: Calculate "du"

du=___dx

Type answer here

Correct Answer: 8

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Math Response

$$\int8x\cdot\cos\left(2x\right)dx$$

Step 2.2: Calculate "v"

(you may have to use U-Substitution)

v=___

Type answer here

Correct Answer: \frac{\sin\left(2x\right)}{2}

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Multiple Choice

$$uv-\int vdu$$

Step 3.1: Substitute in the formula

• $$\left(8x\right)\left(\frac{\sin\left(2x\right)}{2}\right)-\int\left(\frac{\sin\left(2x\right)}{2}\right)\left(8dx\right)$$

• $$\left(8\right)\left(\frac{\sin\left(2x\right)}{2}\right)-\int\left(\frac{\sin\left(2x\right)}{2}\right)\left(8xdx\right)$$

Multiple Choice

$$\left(8x\right)\left(\frac{\sin\left(2x\right)}{2}\right)-\int\left(\frac{\sin\left(2x\right)}{2}\right)\left(8dx\right)$$

Choose the correct answer:

Step 3.2: Simplification

• $$4x\cdot\sin\left(2x\right)-4\int\sin\left(2x\right)dx$$

• $$4x\cdot\sin\left(2x\right)+4x\int\cos\left(2x\right)dx$$

Multiple Choice

$$4x\cdot\sin\left(2x\right)-4\int\sin\left(2x\right)dx$$

Choose the correct answer:

Step 3.3: Solving the remaining integral

(you may have to use U-Substitution)

• $$4x\cdot\sin\left(2x\right)-4\left(-\frac{\cos\left(2x\right)}{2}+C\right)$$

• $$4x\cdot\sin\left(2x\right)-4\left(\frac{\cos\left(2x\right)}{2}+C\right)$$

Math Response

$$4x\cdot\sin\left(2x\right)-4\left(-\frac{\cos\left(2x\right)}{2}+C\right)$$

Step 3.4: Write down the final answer

$$\int8x\cdot\cos\left(2x\right)dx=$$ _____+C

Type answer here

Correct Answer: 4x\times\sin\left(2x\right)+2\cos\left(2x\right)

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