Q 50. Solve the integral:聽鈭玿3e-xdx.... [FREE SOLUTION]

91影视

Chapter 5: Q 50. (page 429)

Solve the integral: $鈭� x^{3} e^{- x} d x$ .

Short Answer

Expert verified

The required answer is $- \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} - \frac{6 x}{e^{x}} - 6 e^{- x} + c$ .

Step by step solution

Step 1. Given information.

We have given integral is

$鈭� x^{3} e^{- x} d x = 鈭� \frac{x^{3}}{e^{x}} d x$ .

Step 2. Solve the integration by parts .

We have,

$u = x^{3} d u = 3 x^{2} d x$

and

$d v = \frac{d x}{e^{x}} v = 鈭� \frac{1}{e^{x}} d u v = - \frac{1}{e^{x}}$

The formula of integration by parts is $鈭� u d v = u v - 鈭� v d u$

$鈭� \frac{x^{3}}{e^{x}} d x = (x^{3} (- \frac{1}{e^{x}}) - 鈭� (- \frac{1}{e^{x}}) 3 x^{2} d x) = - \frac{x^{3}}{e^{x}} - 鈭� (- \frac{3 x^{2}}{e^{x}}) d x = - \frac{x^{3}}{e^{x}} + 3 鈭� (\frac{x^{2}}{e^{x}}) d x$

Step 3. Integration by parts.

We have,

$u = x^{2} d u = 2 x d x$

and

$d v = \frac{d x}{e^{x}} v = 鈭� \frac{1}{e^{x}} d u v = - \frac{1}{e^{x}}$

So,

$- \frac{x^{3}}{e^{x}} + 3 (x^{2} (- \frac{1}{e^{x}}) - 鈭� 2 x (- \frac{1}{e^{x}}) d x) = - \frac{x^{3}}{e^{x}} + 3 (- \frac{x^{2}}{e^{x}} + 2 鈭� (\frac{x}{e^{x}}) d x) = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 鈭� \frac{x}{e^{x}} d x$

Step 4. Integration by parts.

We have,

$u = x d u = d x$

and

$d v = \frac{d x}{e^{x}} v = 鈭� \frac{1}{e^{x}} d u v = - \frac{1}{e^{x}}$

So,

role="math" localid="1648804483219" $- \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 鈭� \frac{x}{e^{x}} d x = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 (- \frac{x}{e^{x}} - 鈭� (- \frac{1}{e^{x}}) d x) = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 (- \frac{x}{e^{x}} + 鈭� (\frac{1}{e^{x}}) d x) = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} + 6 (- \frac{x}{e^{x}} + 鈭� e^{- x} d x) = - \frac{x^{3}}{e^{x}} - \frac{3 x^{2}}{e^{x}} - \frac{6 x}{e^{x}} - 6 e^{- x} + c$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve the integral: $鈭� x^{3} e^{- x} d x$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

Step 3. Integration by parts.

Step 4. Integration by parts.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Decision Maths

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Solve the integral:鈭�x3e-xdx.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

Step 3. Integration by parts.

Step 4. Integration by parts.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Decision Maths

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Solve the integral: $鈭� x^{3} e^{- x} d x$ .