Integrating problems by hand

Question

Integrate.

Do this problem by hand (do not use the Table of Integrals).

Show all of your steps.

∫5 / (64+𝑥2) 𝑑𝑥=  ?  + C

Which strategy did you choose? 

u-Substitution

Partial Fractions

Trig substitution

integration by parts

algebraic manipulation / direct integration

Answer 1

use an integral table, Integral of du/(a^2 +u^2) = (1/a)arctan(u/a) + C

Integral of 5/(64+x^2)dx

= 5 times Integral of dx/(64+x^2)

= 5(1/sqr64)arctan(x/sqr64) + C

=(5/8)arctan(x/8) + C

or use trig substitution if tables are off the table

use x = 8tanT, and dx= 8sec^2(T)dT

which leads to the above arctangent term

if the problem had a negative sign in the denominator

integral of 5dx/(64-x^2), then use x=8sinT, dx=8cosT dT

which leads to a natural log term as the answer

still the integral tables provide a check on your solution

if you solve a problem two different ways and get the same answer

that's nearly certain that's the correct answer

Answer 2

Use Trig Substitution with x = 8tanθ. Then dx = 8sec²θdθ.

5 ∫[dx / (64 + x²)] = 5 ∫ [8sec²θ / (64 + 64tan²θ]dθ = (5/8) ∫ dθ = (5/8)θ + C = (5/8)Arctan(x/8) + C