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Fill in the blanks to complete each of the following integration formulas.

tanxdx=______

Short Answer

Expert verified

The obtained result is-lncosx+C.

Step by step solution

01

Step 1. Given information.

Consider the given integration.

tanxdx

02

Step 2. Find the integration.

Simplify the expression and find the integration.

tanxdx=sinxcosxdx

Put cosx=tand dt=-sinxdx.

Substitute the above values in integration.

tanxdx=-1tdt=-lnt+C=-lncosx+C

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Most popular questions from this chapter

For each function u(x) in Exercises 9–12, write the differential du in terms of the differential dx.

u(x)=x2+1

Consider the integral 1x21x2dxfrom the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution u=sin1xto solve this integral.

(b) Use the trigonometric substitution x=sinu to solve the integral.

(c) Compare and contrast the two methods used in parts (a) and (b).

Solve each of the integrals in Exercises 39–74. Some integrals require trigonometric substitution, and some do not. Write your answers as algebraic functions whenever possible.

xx2+1dx

True/False: Determinewhethereachofthestatementsthat follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: f(x)=x+1x-1is a proper rational function.

(b) True or False: Every improper rational function can be expressed as the sum of a polynomial and a proper rational function.

(c) True or False: After polynomial long division of p(x) by q(x), the remainder r(x) has a degree strictly less than the degree of q(x).

(d) True or False: Polynomial long division can be used to divide two polynomials of the same degree.

(e) True or False: If a rational function is improper, then polynomial long division must be applied before using the method of partial fractions.

(f) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Ax2+Bx-3

(g) True or False: The partial-fraction decomposition of x2+1x2(x-3)is of the form Bx+Cx2+Ax-3.

(h) True or False: Every quadratic function can be written in the formA(x-k)2+C

Explain why 2xx2+1dxand 1xlnxdxare essentially the same integral after a change of variables.

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