(a) Compute the indefinite integral (find the antiderivatives) of| e3* cos(4x) dx.(b) Write down the complete set of parts required to evaluate the above indefiniteintegral by integration by parts.(c) Write down the general formula or just the formula mumber of the Stewart's tableof integration that corresponds to the above indefinite integral. (Note that the tableof integration provided on huskyct follows the same mumbering of the Stewart's book.)(d) Apply the Fundamental Theorem of Calculus to evaluatee ax cos(4x) dx.

(a) We know that ∫eaxcosbx+cdx=eaxa2+b2acosbx+c+bsinbx+c+C. Now putting a=3, b=4 and c=0, we get…

Answered: (a) Compute the indefinite integral…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

(a) Compute the indefinite integral (find the antiderivatives) of
| e3* cos(4x) dx.
(b) Write down the complete set of parts required to evaluate the above indefinite
integral by integration by parts.
(c) Write down the general formula or just the formula mumber of the Stewart's table
of integration that corresponds to the above indefinite integral. (Note that the table
of integration provided on huskyct follows the same mumbering of the Stewart's book.)
(d) Apply the Fundamental Theorem of Calculus to evaluate
e ax cos(4x) dx.

Expert Solution

Step 1

(a) We know that $\int e^{a x} \cos (b x + c) d x = \frac{e^{a x}}{a^{2} + b^{2}} [a \cos (b x + c) + b \sin (b x + c)] + C$ .

Now putting a=3, b=4 and c=0, we get anti-derivative or indefinite integral of the given integral such as

$\int e^{3 x} \cos (4 x) d x = \frac{e^{3 x}}{25} [3 \cos (4 x) + 4 \sin (4 x)] + C$

(b) Let $I = \int e^{3 x} \cos (4 x) d x$ , now by Integration by parts, we get

$I = \int e^{3 x} \cos (4 x) d x = e^{3 x} (\frac{\sin (4 x)}{4}) - \int 3 e^{3 x} (\frac{\sin (4 x)}{4}) d x = e^{3 x} (\frac{\sin (4 x)}{4}) - \frac{3}{4} \int e^{3 x} \sin (4 x) d x = e^{3 x} (\frac{\sin (4 x)}{4}) - \frac{3}{4} [e^{3 x} (\frac{- \cos (4 x)}{4}) - \int 3 e^{3 x} (\frac{- \cos (4 x)}{4}) d x] = e^{3 x} (\frac{\sin (4 x)}{4}) + \frac{3}{16} e^{3 x} \cos (4 x) - \frac{9}{16} \int e^{3 x} \cos (4 x) d x \Rightarrow I = = e^{3 x} (\frac{\sin (4 x)}{4}) + \frac{3}{16} e^{3 x} \cos (4 x) - \frac{9}{16} I \Rightarrow (1 + \frac{9}{16}) I = e^{3 x} (\frac{\sin (4 x)}{4}) + \frac{3}{16} e^{3 x} \cos (4 x) \Rightarrow \frac{25}{16} I = \frac{e^{3 x}}{16} [4 \sin (4 x) + 3 \cos (4 x)] \Rightarrow I = \frac{e^{3 x}}{25} [4 \sin (4 x) + 3 \cos (4 x)]$

Hence, $\int e^{3 x} \cos (4 x) d x = \frac{e^{3 x}}{25} [4 \sin (4 x) + 3 \cos (4 x)] + C$

Step by step

Solved in 2 steps

Knowledge Booster

$Indefinite Integral$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated