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Chapter 4: Q. 50 (page 315)

Indefinite integrals of combinations: Fill in the blanks to complete the integration rules that follow. You may assume that f and g are continuous functions and that k is any real number.

鈭�kf(x)dx=...

Short Answer

Expert verified

The complete integration rule is 鈭�kf(x)dx=k鈭�f(x)dx.

Step by step solution

01

Step 1. Given information.

Consider the given indefinite integral is 鈭�kf(x)dx.

02

Step 2. Fill the blank for the indefinite integral ∫kf(x)dx.

The theorem of Constant Multiple for Indefinite Integrals states that,鈭�kf(x)dx=k鈭�f(x)dx.

Therefore, it can be said the appropriate fill in black will be鈭�kf(x)dx=k鈭�f(x)dx.

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Most popular questions from this chapter

Show by exhibiting a counterexample that, in general, 鈭�f(x)g(x)dx鈮�鈭�f(x)dx鈭�g(x)dx. In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

Compare the definitions of the definite and indefinite integrals. List at least three things that are different about these mathematical objects.

Use a sentence to describe what the notation鈭�k=2100k means. (Hint: Start with 鈥淭he sum of....鈥�)

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

鈭�鈭�11鈥�2x4xdx

Show thatF(x)=12ex2x2-1is an anti-derivative off(x)=x3ex2.
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