How to solve this integral without using substitution method? As I'm curious that is there another method to solve integral? I did integration by parts.
$$\int \sqrt{4-x^2} dx=x\sqrt{4-x^2}+\int \frac{x^2}{\sqrt{4-x^2}}dx$$
$$=x\sqrt{4-x^2}-\int \frac{4-x^2}{\sqrt{4-x^2}}dx+\int \frac{4}{\sqrt{4-x^2}}dx$$
$$=x\sqrt{4-x^2}-\int \sqrt{4-x^2}dx+4\sin ^{-1}(\frac{x}{2})$$
$$2\int\sqrt{4-x^2}dx=x\sqrt{4-x^2}+4\sin ^{-1}(\frac{x}{2})$$
$$\int \sqrt{4-x^2}dx=\frac{1}{2}x\sqrt{4-x^2}+2\sin ^{-1}(\frac{x}{2})$$
Is there any other method can solve integral other than substitution and this? I think Riemann Sum also can be used to solved. But people riemann sum is not considered a method of integration. I wonder why. Thanks a lot.
No comments:
Post a Comment