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Trigonometric integrals: The integrals that follow can be solved by using algebra to write the integrands in the form f'(u(x))u'(x)so thatu-substitution will apply.

Solve sec4xtan3xdxby using the Pythagorean identity role="math" localid="1649174356968" tan2x+1=sec2xto rewrite the integrand as (tan2x+1)tan3xsec2x and then applying substitution with u=tanx.

Short Answer

Expert verified

The value of integral is sec4xtan3xdx=sec6x6+sec4x4+C.

Step by step solution

01

Step 1. Given Information 

Solve sec4xtan3xdxby using the Pythagorean identity tan2x+1=sec2xto rewrite the integrand as (tan2x+1)tan3xsec2x and then applying substitution with u=tanx.

02

Step 2. The given integral is ∫sec4xtan3xdx

We can write as

sec4xtan3xdx=(tan2x+1)tan3xsec2xdx

Let

u=tanxdudx=sec2xdu=sec2xdx

03

Step 3. Now the integral is 

sec4xtan3xdx=(u2+1)u3dusec4xtan3xdx=(u2·u3+1·u3)dusec4xtan3xdx=(u5+u3)dusec4xtan3xdx=u5du+u3dusec4xtan3xdx=u5+15+1+u3+13+1+Csec4xtan3xdx=u66+u44+Csec4xtan3xdx=sec6x6+sec4x4+C

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Most popular questions from this chapter

Problem Zero: Read the section and make your own summary of the material.

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 the integral:x2e3xdx.

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.

13-x2dx

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

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