2011_Tutorial_3 - Z x sin x 2 dx(b Z t 2 e 4 t dt(c Z...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 3 1. Evaluate the following definite integrals. (a) Z 2 1 s 2 + s s 2 ds . (b) Z 4 - 4 | x | dx . (c) Z π 0 1 2 (cos x + | cos x | ) dx . (d) Z π 0 sin 2 1 + θ 2 · dθ . Ans . (a) 1 + 2 - 2 3 / 4 (b) 16 (c) 1 (d) 1 2 π + sin 2 2. Using the fundamental theorem of Calculus, find the derivative dy/dx for the following functions. (a) y = Z x 0 cos t dt . (b) y = Z x 2 0 cos t dt . (c) y = Z sin x 0 dt 1 - t 2 , | x | < π 2 . Ans . (a) cos x 2 x (b) 2 x cos x (c) 1 3. Using the substitution method, or otherwise, find the following integrals. (a) Z x 1 / 2 sin( x 3 / 2 + 1) dx . (b) Z csc 2 2 t cot 2 t dt . (c) Z 1 θ 2 sin 1 θ cos 1 θ dθ . (d) Z 18 tan 2 x sec 2 x (2 + tan 3 x ) dx . (e) Z sin θ θ cos 3 θ . Ans . (a) - 2 3 cos( x 3 / 2 + 1) + C (b) - 1 4 cot 2 2 t + C
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MA1505 Tutorial 3 (c) - 1 2 sin 2 1 θ + C (d) 6 ln | tan 3 x + 2 | + C (e) sec 2 θ + C 4. Applying the method of integration by parts , or otherwise, find the following integrals.
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Unformatted text preview: Z x sin ‡ x 2 · dx. (b) Z t 2 e 4 t dt. (c) Z e-y cos y dy . (d) Z θ 2 sin(2 θ ) dθ . (e) Z z (ln z ) 2 dz . (f) Z { sin e-x + e x cos e-x } dx . Ans . (a)-2 £ x cos ( x 2 )-2sin ( x 2 )/ + C (b) ( t 2 4-t 8 + 1 32 ) e 4 t + C (c) e-y 2 (sin y-cos y ) + C (d)-1 2 £ θ 2 cos(2 θ )-θ sin(2 θ )-1 2 cos(2 θ ) / + C (e) 1 2 h z 2 (ln z ) 2-z 2 (ln z ) + z 2 2 i + C (f) e x cos e-x + C 5. Suppose f : R +-→ R is a continuous function. Show that Z π xf (sin x ) dx = π 2 Z π f (sin x ) dx. ( Hint: Use the substitution u = π-x . ) Hence find Z π x sin 4 x dx . Ans . 3 16 π 2 . 2...
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