1 Functions Defined in Terms of Integrals


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1 November 5, 8 MAT86 Week 3 Justin Ko Functions Defined in Terms of Integrls Integrls llow us to define new functions in terms of the bsic functions introduced in Week. Given continuous function f(), consider the re function F (). Generl Properties of F (): The following properties will llow us to sketch F () even if the definite integrl is impossible to simplify:. F () is continuous where it is defined. (Fundmentl theorem of clculus). F (). (Definition of the definite integrl) 3. F () f(). (Fundmentl theorem of clculus) 4. F () f (). (Fundmentl theorem of clculus) 5. If nd f() is even, then F () is odd. (Chnge of vribles) 6. If f() is odd, then F () is even. (Chnge of vribles). The Nturl Logrithm Definition. For >, the nturl logrithm is defined by ln() t dt. Sketching the Curve: Using the bsic properties of integrl defined functions for F () ln() we know tht:. The intercepts nd derivtives of F re given by F (), F (), F ().. With some work, we cn lso show tht lim + F () nd lim F (). Therefore, we cn conclude tht F () is strictly incresing concve down function tht psses through the point (, ). The second point lso tells us there is verticl symptote t nd the integrl diverges to s. f() / F () ln() Figure : The grph of f() nd F () ln() re displyed bove. The vlue of F () is the re under the curve of ln() between nd. Pge of 5
2 November 5, 8 MAT86 Week 3 Justin Ko. The Error Function Definition. For R, the error function is defined by erf() e t dt. Sketching the Curve: Using the bsic properties of integrl defined functions for F () erf() we know tht. The intercepts nd derivtives of F re given by F (), F () e, F () 4 e.. With some work, we cn lso show tht lim F (). 3. F () is odd since e is even. Therefore, we cn conclude for tht F () is strictly incresing concve down function tht psses through the point (, ). The second point lso implies tht y is horizontl symptote s. Since F () is odd, we cn recover the shpe for < by reflecting round the origin. f() e F () erf() Figure : The grph of f() e nd F () erf() re displyed bove. The vlue of F () is the re under the curve of e between nd..3 Emple Problems.3. Properties About Integrl Defined Functions Problem. ( ) Let Show tht F () f() nd F () f (). F (). Solution. By the first prt of the fundmentl theorem of clculus, Differentiting this gin implies F () d d F () f (). f(). Pge of 5
3 November 5, 8 MAT86 Week 3 Justin Ko Problem. ( ) Suppose f() is even (f( ) f()). Show tht the function is n odd function. F () Solution. It suffices to show F ( ) F (). Using the chnge of vribles u t, we hve du dt, t u, t u F ( ) F (). f( u) du f( u) f(u) Problem 3. ( ) Suppose f() is odd (f( ) f()). Show tht the function is n even function. F () Solution 3. It suffices to show F ( ) F (). Using the chnge of vribles u t, we hve du dt, t u, t u F ( ) f( u) du. f( u) f(u) It my pper tht the lst term is not of the sme form s the term F () becuse the lower bounds of integrtion re different. However, we cn split the region of integrtion nd use chnge of vribles to conclude tht + f( ũ) dũ + f(ũ) dũ + F (). ũ u, dũ du, du f( u) f(u) dũ Pge 3 of 5
4 November 5, 8 MAT86 Week 3 Justin Ko.3. The Nturl Logrithm Problem. ( ) Using the integrl definition of the nturl logrithm, show tht ln() d ln() + C. Solution. We cn integrte by prts to recover the formul for the ntiderivtive, ± D I + ln() d d ln(). Since d d ln() by the fundmentl theorem, we hve ln() d ln() d ln() + C. Remrk: It is esy to check tht the ln() + C is n ntiderivtive by simply differentiting. Problem. ( ) Using the integrl definition of the nturl logrithm, show tht ln(y) ln() + ln(y) Solution. We wnt to write ln(y) in terms of its integrl definition. The trick is to split the integrl ln(y) y t dt t dt + t dt + y y ln() + ln(y). t dt u du b c u t dt, du, + y dt b c y du.3.3 The Error Function Problem. ( ) Using the integrl definition of the error function, show tht erf() d erf() + e + C. Solution. We cn integrte by prts to recover the formul for the ntiderivtive, ± D I + erf() d d erf() Pge 4 of 5
5 November 5, 8 MAT86 Week 3 Justin Ko Since d d erf() e by the fundmentl theorem, we hve erf() d erf() e d. The second integrl cn be solved using the substitution u, du d which gives us erf() d erf() + e u du erf() + e + C Remrk: It is esy to check tht the erf() + e + C is n ntiderivtive by simply differentiting. Problem. ( ) Using the integrl definition of the error function, show tht e t e t dt e 4 erf ) ( )) + erf. Solution. We wnt to write the integrl in terms of the error function. The trick is to complete the squre in the eponent e t e t dt e 4 e 4 e 4 e t +t dt (complete the squre) ( ( e (t ) dt e 4 e e e u du e u du e u du + ) b e u du u t, du dt, dt ) e u du erf ) ( erf )) e t dt erf ) ( )) + erf. (erf() is odd) + erf() b du Pge 5 of 5
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