## Part - 3 Lecture - 3 Chapter 7 Integrals

The Way Get Started Is To Quit Talking And Begin Doing.” – Walt Disney

Booklets/Notes/Assignments are typed here on this website and their PDFs will be made available soon.

:::INTEGRATION BY PARTS:::

Method to identify the First Function:

I    L    A    T    E

I = Inverse Trigonometric Functions
L = Logarithmic Functions
A = Algebraic Functions
T = Trigonometric Functions
E = Exponential functions

There’s a special case $$\int{e^x(f(x)+f^\prime(x))=e^xf(x)+C}$$. You may need a little bit of simplification to apply this identity:

12. $$\int{e^x(\sin{x}+\cos{x})dx= e^x\sin{x}}+C$$

13. $$\int\frac{xe^x}{(1+x)^2}dx=\frac{e^x}{1+x}+C$$

14. $$\int\frac{(x-4)e^x}{(x-2)^3}dx=\frac{e^x}{(x-2)^2}+C$$

15. $$\int\frac{2+\sin{2}x}{1+\cos{2}x}e^xdx= e^x\tan{x}+C$$

16. $$\int{e^x\left(\frac{\sin{4}x-4}{1-\cos{4}x}\right)}dx= e^x.\cot{2}x+C$$

17. $$\int\frac{(x^2+1)e^x}{(x+1)^2}dx=\left(\frac{x-1}{x+1}\right)e^x+C$$

18. $$\int{e^x\left(\frac{1-x}{1+x^2}\right)^2dx=\frac{e^x}{1+x^2}}+C$$

19. $$\int_{\frac{\pi}{2}}^{\pi}{e^x\left(\frac{1-\sin{x}}{1-\cos{x}}\right)}dx= e^\frac{\pi}{2}$$