121616949-math.174 - tution and trigonometric identities...

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160 Chapter 8 Techniques of Integration Exercises Find the antiderivatives. 1. Z (1 - t ) 9 dt 2. Z ( x 2 + 1) 2 dx 3. Z x ( x 2 + 1) 100 dx 4. Z 1 3 1 - 5 t dt 5. Z sin 3 x cos x dx 6. Z x p 100 - x 2 dx 7. Z x 2 1 - x 3 dx 8. Z cos( πt ) cos sin( πt ) dt 9. Z sin x cos 3 x dx 10. Z tan x dx 11. Z π 0 sin 5 (3 x ) cos(3 x ) dx 12. Z sec 2 x tan x dx 13. Z π/ 2 0 x sec 2 ( x 2 ) tan( x 2 ) dx 14. Z sin(tan x ) cos 2 x dx 15. Z 4 3 1 (3 x - 7) 2 dx 16. Z π/ 6 0 (cos 2 x - sin 2 x ) dx 17. Z 6 x ( x 2 - 7) 1 / 9 dx 18. Z 1 - 1 (2 x 3 - 1)( x 4 - 2 x ) 6 dx 19. Z 1 - 1 sin 7 x dx 20. Z f ( x ) f ( x ) dx Functions consisting of products of the sine and cosine can be integrated by using substi-
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Unformatted text preview: tution and trigonometric identities. These can sometimes be tedious, but the technique is straightforward. Some examples will suffice to explain the approach. EXAMPLE 8.5 Evaluate Z sin 5 x dx . Rewrite the function: Z sin 5 x dx = Z sin x sin 4 x dx = Z sin x (sin 2 x ) 2 dx = Z sin x (1-cos 2 x ) 2 dx....
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