Explanation Given Information: The provided integral is ∫ ( x e − x 2 + 1 + e 2 x ) d x . Formula used: Follow the steps as mentioned below to solve an integral ∫ f d x by using substitution rule. Step 1 : Assume a function u in terms of x from f . Step 2 : Take the derivative d u d x of the function obtained in step 1 and solve for d x in terms of d u . Step 3 : Substitute the value of u from step 1 and the value of d x from step 2 in the provided integral and then solve the integral. The sum for an integral, ∫ [ f ( x ) + g ( x ) ] d x = ∫ f ( x ) d x + ∫ g ( x ) d x Where f ( x ) and g ( x ) are two functions The constant multiple rule, ∫ k ⋅ f ( x ) d x = k ∫ f ( x ) d x Where k is a constant. The rule for the indefinite integral of the exponential e x , ∫ e x d x = e x + C Where C is the constant of integration. Calculation: Consider the integral, ∫ ( x e − x 2 + 1 + e 2 x ) d x To evaluate the integral ∫ ( x e − x 2 + 1 + e 2 x ) d x apply sum for an integral, ∫ ( x e − x 2 + 1 + e 2 x ) d x = ∫ x e − x 2 + 1 d x + ∫ e 2 x d x Integrate individual terms of the expression ∫ x e − x 2 + 1 d x + ∫ e 2 x d x and then combine the result. Integrate first term ∫ x e − x 2 + 1 d x by using substitution rule as, Assume a function of u in terms of x from x e − x 2 + 1 , u = − x 2 + 1 Take the derivative d u d x of the function u = − x 2 + 1 , d u d x = − 2 x Solve for d x , d x = − 1 2 x d u Now substitute u = − x 2 + 1 and d x = − 1 2 x d u in the integral term ∫ x e − x 2 + 1 d x , this gives, ∫ x e − x 2 + 1 d x = ∫ x ⋅ e u ⋅ ( − 1 2 x ) d u = ∫ − x 2 ⋅ e u ⋅ 1 x d u Simplify the integral by cancelling x by x . ∫ − x 2 ⋅ e u ⋅ 1 x d u = ∫ − e u 2 d u Evaluate the integral ∫ − e u 2 d u by using constant multiple rule and rule for the indefinite integral of the exponential e x , ∫ − e u 2 d u = − 1 2 ∫ e u d u = − 1 2 e u + C Therefore, the solution of the integral ∫ − 1 2 e u d u is − 1 2 e u + C

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 13.2, Problem 52E

To determine

To calculate: The solution of the integral ∫(xe−x2+1+e2x)dx.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.2, Problem 51E

Chapter 13.2, Problem 53E

Students have asked these similar questions

Does the series converge or diverge

Diverge or conver

Chapter 13 Solutions

Finite Mathematics and Applied Calculus (MindTap Course List)

Ch. 13.1 - In Exercises 142, evaluate the integral. [HinT:...Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The solution of the integral ∫ ( x e − x 2 + 1 + e 2 x ) d x .

Solution Summary: The author explains how to solve an integral using substitution rule.

Videos

To calculate: The solution of the integral ∫(xe−x2+1+e2x)dx.

Want to see the full answer?

Chapter 13 Solutions