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Chapter 4: Q. 14 (page 362)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

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

Step 2. Verification.

We can write the differentiation as,

$\frac{d}{d x} \ln (\sin x) = \frac{1}{\sin x} (\cos x)$

$= \frac{\cos x}{\sin x} = c o t x$

Hence, $\ln (\sin x) is an anti-derivative of \cot x .$

Therefore,

$\int \cot x d x = \ln (\sin x) + C$

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Most popular questions from this chapter

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

$\int (x^{2} + 2^{x} + 2^{2}) d x$

Sum and constant-multiple rules: State the sum and constant-multiple rules for (a) derivatives and (b) limits.

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

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

Given a simple proof that if n is a positive integer and c is any real number, then $\sum_{k = 1}^{n} c = c n$

Calculate the exact value of each definite integral in Exercises 47–52 by using properties of definite integrals and the formulas in Theorem 4.13.

$\int_{2}^{4} (x^{2} + 1) d x$

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Verify that∫cotxdx=ln(sinx)+C. (Do not try to solve the integral from scratch.)

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Verification.

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Most popular questions from this chapter

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