Math trip 1912 3 integrals containing the exponential

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( Math. Trip. 1912.) 3. Integrals containing the exponential function. Prove that Z e ax cos bx dx = a cos bx + b sin bx a 2 + b 2 e ax , Z e ax sin bx dx = a sin bx - b cos bx a 2 + b 2 e ax . [Denoting the two integrals by I , J , and integrating by parts, we obtain aI = e ax cos bx + bJ, aJ = e ax sin bx - bI. Solve these equations for I and J .] 4. Prove that the successive areas bounded by the curve of Ex. 2 and the positive half of the axis of x form a geometrical progression, and that their sum is b a 2 + b 2 1 + e - aπ/b 1 - e - aπ/b . 5. Prove that if a > 0 then Z 0 e - ax cos bx dx = a a 2 + b 2 , Z 0 e - ax sin bx dx = b a 2 + b 2 . 6. If I n = Z e ax x n dx then aI n = e ax x n - nI n - 1 . [Integrate by parts. It follows that I n can be calculated for all positive integral values of n .]
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[IX : 210] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 462 7. Prove that, if n is a positive integer, then Z ξ 0 e - x x n dx = n ! e - ξ e ξ - 1 - ξ - ξ 2 2! - · · · - ξ n n ! and Z 0 e - x x n dx = n ! . 8. Show how to find the integral of any rational function of e x . [Put x = log u , when e x = u , dx/du = 1 /u , and the integral is transformed into that of a rational function of u .] 9. Integrate e 2 x ( c 2 e x + a 2 e - x )( c 2 e x + b 2 e - x ) , distinguishing the cases in which a is and is not equal to b . 10. Prove that we can integrate any function of the form P ( x, e ax , e bx , . . . ), where P denotes a polynomial. [This follows from the fact that P can be ex- pressed as the sum of a number of terms of the type Ax m e kx , where m is a positive integer.] 11. Show how to integrate any function of the form P ( x, e ax , e bx , . . . , cos lx, cos mx, . . . , sin lx, sin mx, . . . ) . 12. Prove that Z a e - λx R ( x ) dx , where λ > 0 and a is greater than the greatest root of the denominator of R ( x ), is convergent. [This follows from the fact that e λx tends to infinity more rapidly than any power of x .] 13. Prove that Z -∞ e - λx 2 + μx dx , where λ > 0, is convergent for all values of μ , and that the same is true of Z -∞ e - λx 2 + μx x n dx , where n is any positive integer. 14. Draw the graphs of e x 2 , e - x 2 , xe x , xe - x , xe x 2 , xe - x 2 , and x log x , de- termining any maxima and minima of the functions and any points of inflexion on their graphs. 15. Show that the equation e ax = bx , where a and b are positive, has two real roots, one, or none, according as b > ae , b = ae , or b < ae . [The tangent to
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[IX : 210] THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS 463 the curve y = e ax at the point ( ξ, e ) is y - e = ae ( x - ξ ) , which passes through the origin if = 1, so that the line y = aex touches the curve at the point (1 /a, e ). The result now becomes obvious when we draw the line y = bx . The reader should discuss the cases in which a or b or both are negative.] 16. Show that the equation e x = 1 + x has no real root except x = 0, and that e x = 1 + x + 1 2 x 2 has three real roots. 17. Draw the graphs of the functions log( x + p x 2 + 1) , log 1 + x 1 - x , e - ax cos 2 bx, e - (1 /x ) 2 , e - (1 /x ) 2 p 1 /x, e - cot x , e - cot 2 x .
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