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Solve each of the definite integrals in Exercises 67–76.

0π/4sin2xcos4xdx.

Short Answer

Expert verified

The answer is13.

Step by step solution

01

Step 1. Given Information.

The integral is0π/4sin2xcos4xdx.

02

Step 2. Explanation.

Simplify the integral using trigonometric identities, tanx=sinxcosxand secx=1cosx.

0π/4tan2xsec2xdx

Substituteu=tanxso, du=sec2xdx

so, limits changes 0 to 1.

role="math" localid="1649243192222" 01u2du.

03

Step 3. Calculation.

Further simplify.

01u2du=u3301=13

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Most popular questions from this chapter

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.

x3x2+1dx

Suppose you use polynomial long division to divide p(x) by q(x), and after doing your calculations you end up with the polynomial x2-x+3 as the quotient above the top line, and the polynomial 3x − 1 at the bottom as the remainder. Thenp(x)=___andp(x)q(x)=____

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingx2+45/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2a2.

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

Solve given definite integrals.

04xx2+4dx

Solve the integralx3x2-1dxthree ways:

(a) with the substitution u=x2-1,followed by back substitution;

(b) with integration by parts, choosing localid="1648814744993" u=x2anddv=xx2-1dx;

(c) with the trigonometric substitution x = sec u.

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