Chapter 5: Q 36. (page 429)

Solve the integral: $\int \frac{x^{2}}{e^{x}} d x$

Short Answer

Expert verified

The required answer is $(- \frac{x}{e^{x}}) - \frac{x}{2 e^{x}} - \frac{1}{2} e^{- x} + c$ .

Step by step solution

Step 1. Given information.

We have given integral is $\int \frac{x^{2}}{e^{x}} d x$ .

Step 2. Solve the integration by parts .

We have, $u = x^{2}, d v = \frac{d x}{e^{x}}$

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

and

$d v = \frac{d x}{e^{x}} v = \int \frac{d x}{e^{x}} v = - \frac{1}{e^{x}}$

The formula of integration by parts is $\int u d v = u v - \int v d u$ .

role="math" localid="1648713324601" $\int \frac{x^{2}}{e^{x}} d x = [x^{2} (- \frac{1}{e^{x}}) - \frac{1}{2} \int x (- \frac{1}{e^{x}}) d x] = (- \frac{x}{e^{x}}) + \frac{1}{2} \int \frac{x}{e^{x}} d x = (- \frac{x}{e^{x}}) + \frac{1}{2} [x (- \frac{1}{e^{x}}) - \int (- \frac{1}{e^{x}}) d x] = (- \frac{x}{e^{x}}) + \frac{1}{2} [(- \frac{x}{e^{x}}) + \int e^{- x} d x]$

Step 3. Integration by parts.

We have, $u = - x d u = - d x$

$(- \frac{x}{e^{x}}) + \frac{1}{2} [(- \frac{x}{e^{x}}) + \int e^{- x} d x] = (- \frac{x}{e^{x}}) + \frac{1}{2} [(- \frac{x}{e^{x}}) - e^{- x}] + c = (- \frac{x}{e^{x}}) - \frac{x}{2 e^{x}} - \frac{1}{2} e^{- x} + c$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Solve the integral: $\int \frac{x^{2}}{e^{x}} d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

Step 3. Integration by parts.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

Solve the integral:∫x2exdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integration by parts .

Step 3. Integration by parts.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

Solve the integral: $\int \frac{x^{2}}{e^{x}} d x$