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Chapter 5: Q. 76 (page 465)

Solve each of the integrals in Exercises 75鈥�78by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.
鈭�x2-1x2+1dx

Short Answer

Expert verified

The value of integral isx-2tan-1(x)+C

Step by step solution

01

Step 1. Given informaion

An expression is given as鈭�x2-1x2+1dx

02

Step 2. Solving integral

Split the term by using long distance method and simplify as,

x2-1x2+1=1-2x2+1鈭�1-2x2+1dx=鈭�1dx-2鈭�1x2+1dx=x-2tan-1(x)+C

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Most popular questions from this chapter

Consider the integral 鈭�sinxcosxdx.

(a) Solve this integral by using u-substitution with u=sinx and du=cosxdx.

(b) Solve the integral another way, using u-substitution with u=cosx and du=鈭�sinxdx.

(c) How must your two answers be related? Use algebra to prove this relationship.

Show by differentiating (and then using algebra) that 鈭�cot鈦�sin鈭�1鈦�xand 鈭�1鈭�x2xare both antiderivatives of 1x21鈭�x2. How can these two very different-looking functions be an antiderivative of the same function?

Solve the integral:鈭�xexdx

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Solve the integral: 鈭�xsinx2dx.
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