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Chapter 4: Q. 15 (page 362)

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

Short Answer

Expert verified

It is verified that $\int \ln x d x = x (\ln x - 1) + C .$

Step by step solution

01

Step 1. Given information.

The given integral is $\int \ln x d x = x (\ln x - 1) + C .$

02

Step 2. Verification.

Now, we can write,

$\frac{d}{d x} x (\ln x - 1) = 1 (\ln x - 1) + x (\frac{1}{x} - 0) = \ln x - 1 + 1 = \ln x$

So,

$x (\ln x - 1) is an anti-derivative of \ln x .$

Therefore,

$\int \ln x d x = x (\ln x - 1) + C$

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Most popular questions from this chapter

Explain why it would be difficult to write the sum $\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{8} + \frac{1}{11} + \frac{1}{12} + \frac{1}{13}$ in sigma notation.

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.

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

Describe an example that illustrates that $\int_{a}^{b} | f (x) | d x$ is not equal to $| \int_{a}^{b} f (x) d x |$ .

Approximate the same area as earlier but this time with eight rectangles is this over approximation or under approximation of the exact area under the graph

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