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Chapter 4: Q. 11 (page 362)

Write out all the integration formulas and rules that we know at this point.

Short Answer

Expert verified

The formulas are:

$\int x^{k} d x = \frac{1}{k + 1} x^{k + 1} + C \int \frac{1}{x} d x = \ln | x | + C \int e^{k x} d x = \frac{1}{k} e^{k x} + C \int b^{x} d x = \frac{1}{\ln b} b^{x} + C \int \sin x d x = - \cos x + C$

Step by step solution

01

Step 1. Given information.

According to the question, we need to write all the integral formulas.

02

Step 2. Explanation.

The formulas that are known so far are:

$\int x^{k} d x = \frac{1}{k + 1} x^{k + 1} + C {P o w e r f u n c t i o n} \int \frac{1}{x} d x = \ln | x | + C \int e^{k x} d x = \frac{1}{k} e^{k x} + C {E x p o n e n t i a l f u n c t i o n} \int b^{x} d x = \frac{1}{\ln b} b^{x} + C \int \sin x d x = - \cos x + C {T r i g n o m e t r i c F u n c t i o n}$

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Most popular questions from this chapter

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

$\int_{0}^{1} \frac{1}{1 + x^{2}} d x$

Given a simple proof that $\sum_{k = 0}^{n} 3 a_{k} = 3 \sum_{k = 0}^{n} a_{k}$

Prove that $\int_{1}^{3} (3 x + 4) d x = 20$ in three different ways:

(a) algebraically, by calculating a limit of Riemann sums;

(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;

(c) with formulas, by using properties and formulas of definite integrals.

Properties of addition: State the associative law for addition, the commutative law for addition, and the distributive law for multiplication over addition of real numbers. (You may have to think back to a previous algebra course.)

Verify that $\int \ln x d x = x (\ln x - 1) + C$ (Do not try to solve the integral from scratch.

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