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Chapter 5: Q. 44 (page 465)

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

鈭�1x2+432dx

Short Answer

Expert verified

The solution of the integral isx4x2+4+C.

Step by step solution

01

Step 1. Given Information.

The given integral is鈭�1x2+432dx.

02

Step 2. Solve. 

To solve the integral, let x=2sinu, so derivation of uis dx=2cosudu.

Thus, substitute u into the original integral,

role="math" localid="1648802019739" 鈭�1x2+432dx=鈭�12sinu2+4322cosudu=鈭�14sin2u+4322cosudu=鈭�14(sin2u+1)322cosudu=鈭�18sin2u+4322cosudu=14鈭�1sin2u+132cosuduLet'susetheidentitysin2x+cos2x=1=14鈭�1cos2u32cosudu=14鈭�1cos2udu=14tanudu

03

Step 3. Solve. 

By proceeding with the calculation further, substitute back uin the above equation,

=14tansin-1x2du=x4x2+4+C

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