he formula for integration by parts, we have
∫[f(x)g(x)]dx=f(x)∫g(x)dx–∫[ddxf(x)∫g(x)dx]dx
Using the formula above, equation (i) becomes
I=a2–x2−−−−−√∫1dx–∫[ddxa2–x2−−−−−√(∫1dx)]dx⇒I=xa2–x2−−−−−√–∫–x2a2–x2−−−−−√dx⇒I=xa2–x2−−−−−√–∫–a2+a2–x2a2–x2−−−−−√dx⇒I=xa2–x2−−−−−√–∫–a2a2–x2−−−−−√dx–∫a2–x2a2–x2−−−−−√dx⇒I=xa2–x2−−−−−√+a2∫1a2–x2−−−−−√dx–∫a2–x2−−−−−√dx⇒I=xa2–x2−−−−−√+a2sin–1(xa)–I+c⇒I+I=xa2–x2−−−−−√+a2sin–1(xa)+c⇒2I=xa2–x2−−−−−√+a2sin–1(xa)+c⇒I=xa2–x2−−−−−√2+a22sin–1(xa)+c⇒∫a2–x2−−−−−√dx=xa2–x2−−−−−√2+a22sin–1(xa)+c
Integration of the Square Root of a^2+x^2
Integration of x ln x ⇒
Notice that x^4 is (x^2)^2 whose derivative is 2x (which is in the numerator if you multiply and divide the integral by 2). Now it’s basically another integral like the one in the artic
We can write a = (sqrt(a))^2, so (a^2 – x^2) = (sqrt(x^2 – a^2))^2 so this is what we get: (a^2 – x^2) / (sqrt(x^2 – a^2)) = (sqrt(a^2 – x^2))^2 / (sqrt(x^2 – a^2)) = sqrt(x^2 – a^2)
