Question

79. Tabular integration extended Refer to Exercise 77.
a. The following table shows the method of tabular integration applied to
∫ eˣ cos x dx.

Use the table to express ∫ eˣ cos x dx in terms of the sum of functions and an indefinite integral.
b. Solve the equation in part (a) for ∫ eʳ cos z dz.
c. Evaluate ∫ e⁻ᶻ sin 3z dz by applying the idea from parts (a) and (b).

Accepted Answer

Identify the functions from the table: let $$ f(x) = e^x $$ and $$ g(x) = \cos x . $$The table shows derivatives of $$ f $$ and integrals of $$ g $$ along with signs for tabular integration. Use the tabular integration method to write $$ \int e^x \cos x \, dx  as $$the sum of products of $$ f $$ and integrals of $$ g $$ with alternating signs, plus the remaining integral. From the table, this gives:

$$ \int e^x \cos x \, dx = + e^x \sin x - e^x (-\cos x) + \int e^x (-\cos x) \, dx $$ Simplify the expression inside the integral and the sum of functions:

$$ \int e^x \cos x \, dx = e^x \sin x + e^x \cos x - \int e^x \cos x \, dx $$ Recognize that the integral $$ \int e^x \cos x \, dx $$ appears on both sides of the equation. To solve for it, add $$ \int e^x \cos x \, dx  to $$both sides to isolate the integral term. Divide both sides of the resulting equation by 2 to solve explicitly for $$ \int e^x \cos x \, dx . $$This gives the integral in terms of elementary functions without any remaining integral.

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Key Concepts

Tabular Integration Method

Integration by Parts

Solving Integral Equations Involving the Same Integral