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Techniques of Integration EXERCISES

# Techniques of Integration EXERCISES - (8t INTEGRATION BY...

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INTEGRATION BY PARTS Integrate by parts to evaluate the following integrals 1. xe x dx Ans. e x ( x -1) + c. 2. θ cos θ d θ Ans. cos θ + sin θ + c. 3. sin -1 x d x Ans. x sin -1 x + 1 – x 2 + c. 4. ( x 3 + 1) ln x d x Ans. ( 4 1 x 4 + x ) ln x 16 1 x 4 x + c. 5. x 1 x + d x Ans. 15 2 (3 x -2)( x +1) 3/2 +c. 6. dx 4 x x 2 3 + Ans. 15 1 (3 x 2 -8)( x 2 +4) 3/2 +c. 7. x sec 2 3 1 x d x Ans. 3 x tan - 3 1 x + 9ln cos 3 1 x + c. 8. sin (ln x ) d x Ans. 2 1 x [ sin (ln x ) - cos (ln x )] + c. 9. e - x cos 2 x dx Ans. 5 1 e - x (2 sin 2 x – cos 2 x ) + c . 10. t 3 e t dt Ans. e t (t 3 – 3t 2 + 6t – 6) + c . 11. x - - 1 1 1 cos dx Ans. π 12. - 7 0 3 2 3 8 x dx x Ans. 8.7 13. π θ θ θ 2 0 cos 3 sin d Ans. 1/2 14. + 1 0 2 ) 1 ( dx x xe x Ans. 2 1 e - 1 INTEGRATION OF RATIONAL FRACTION Evaluate the following integrals. 1. + x x dx 2 2 Ans. . 2 ln 2 1 c x x + + 2. - + + dx x x x 1 2 2 2 Ans. ( ) dx x x x 1 1 ln 2 + - + 3. -

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Techniques of Integration EXERCISES - (8t INTEGRATION BY...

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