This is in Mathematica in cal 2 Compute the fourth derivative ofcos(x + 1) − ln(x² + 4) + x²²+1with respect to z. Also, compute the integral of the original function on the interval [1, 3]. Do notretype the full expression to be integrated.

Answered: Compute the fourth derivative of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

This is in Mathematica in cal 2

Compute the fourth derivative of
cos(x + 1) − ln(x² + 4) + x²²+1
with respect to z. Also, compute the integral of the original function on the interval [1, 3]. Do not
retype the full expression to be integrated.

Expert Solution

Step 1

Sol:-

First part:-

Fourth Derivative of given function:-

$\frac{d}{d x} (\cos (x + 1) - \ln (x^{2} + 4) + x e^{x^{2} + 1}) = \frac{d}{d x} (\cos (x + 1)) - \frac{d}{d x} (\ln (x^{2} + 4)) + \frac{d}{d x} (x e^{x^{2} + 1}) = - \sin (x + 1) - \frac{2 x}{x^{2} + 4} + e^{x^{2} + 1} + 2 e^{x^{2} + 1} x^{2} A g a i n d i f f e r e n t i a t e : - \frac{d}{d x} (- \sin (x + 1) - \frac{2 x}{x^{2} + 4} + e^{x^{2} + 1} + 2 e^{x^{2} + 1} x^{2}) = - \frac{d}{d x} (\sin (x + 1)) - \frac{d}{d x} (\frac{2 x}{x^{2} + 4}) + \frac{d}{d x} (e^{x^{2} + 1}) + \frac{d}{d x} (2 e^{x^{2} + 1} x^{2}) = - \cos (x + 1) - \frac{2 (- x^{2} + 4)}{{(x^{2} + 4)}^{2}} + e^{x^{2} + 1} \cdot 2 x + 2 (2 e^{x^{2} + 1} x^{3} + 2 e^{x^{2} + 1} x) A g a i n d i f f e r e n t i a t e : - \frac{d}{d x} (\cos (x + 1) - \frac{2 (- x^{2} + 4)}{{(x^{2} + 4)}^{2}} + e^{x^{2} + 1} \cdot 2 x + 2 (2 e^{x^{2} + 1} x^{3} + 2 e^{x^{2} + 1} x)) = \frac{d}{d x} (\cos (x + 1)) - \frac{d}{d x} (\frac{2 (- x^{2} + 4)}{{(x^{2} + 4)}^{2}}) + \frac{d}{d x} (e^{x^{2} + 1} \cdot 2 x) + \frac{d}{d x} (2 (2 e^{x^{2} + 1} x^{3} + 2 e^{x^{2} + 1} x)) - \sin (x + 1) - (- \frac{4 x (- x^{2} + 12)}{{(x^{2} + 4)}^{3}}) + 2 (2 e^{x^{2} + 1} x^{2} + e^{x^{2} + 1}) + 2 (4 e^{x^{2} + 1} x^{4} + 10 e^{x^{2} + 1} x^{2} + 2 e^{x^{2} + 1}) S i m p l i f y : - = 8 e^{x^{2} + 1} x^{4} + 24 e^{x^{2} + 1} x^{2} - \sin (x + 1) + \frac{4 x (- x^{2} + 12)}{{(x^{2} + 4)}^{3}} + 6 e^{x^{2} + 1} A g a i n d i f f e r e n t i a t e : - \frac{d}{d x} (8 e^{x^{2} + 1} x^{4} + 24 e^{x^{2} + 1} x^{2} - \sin (x + 1) + \frac{4 x (- x^{2} + 12)}{{(x^{2} + 4)}^{3}} + 6 e^{x^{2} + 1}) = \frac{d}{d x} (8 e^{x^{2} + 1} x^{4}) + \frac{d}{d x} (24 e^{x^{2} + 1} x^{2}) - \frac{d}{d x} (\sin (x + 1)) + \frac{d}{d x} (\frac{4 x (- x^{2} + 12)}{{(x^{2} + 4)}^{3}}) + \frac{d}{d x} (6 e^{x^{2} + 1}) = 8 (2 e^{x^{2} + 1} x^{5} + 4 e^{x^{2} + 1} x^{3}) + 24 (2 e^{x^{2} + 1} x^{3} + 2 e^{x^{2} + 1} x) - \cos (x + 1) + \frac{12 (x^{4} - 24 x^{2} + 16)}{{(x^{2} + 4)}^{4}} + 12 e^{x^{2} + 1} x = 16 e^{x^{2} + 1} x^{5} + 80 e^{x^{2} + 1} x^{3} + 60 e^{x^{2} + 1} x - \cos (x + 1) + \frac{12 (x^{4} - 24 x^{2} + 16)}{{(x^{2} + 4)}^{4}} H e n c e F o u r t h d e r i v a t i v e i s g i v e n b y : - 16 e^{x^{2} + 1} x^{5} + 80 e^{x^{2} + 1} x^{3} + 60 e^{x^{2} + 1} x - \cos (x + 1) + \frac{12 (x^{4} - 24 x^{2} + 16)}{{(x^{2} + 4)}^{4}}$