# Integration of Exponential and Trignometric functions

• Jan 26th 2010, 01:57 AM
egt2010
Integration of Exponential and Trignometric functions
Let S be the integration symbol

1. S (dx)/ (x. sec ^3 (ln (x)) dx
2. S (x^3)/(sqrt (x^2 -4 )) dx
3. S (2sec x)/(sin x + cos x) dx
4. S sqrt (sqrt (x) -1) dx
5. S sec^3 (x). tan^3 (x) dx
6. S sin (5x)*cos (7x) dx
7. S (e^(x) -3)/(e^(x) +2) dx
8. S (sec^4 (x))/(tan^2 (x)) dx
9. S (sin 2x)/(1 + 3sin^2 (x))^3 dx

Sorry for the many questions, but I wanna make sure of everything related to Integration before my Exam the next Thursday.

• Jan 26th 2010, 02:51 AM
girdav
6. We have for $\displaystyle a\in \mathbb R$ and $\displaystyle b\in \mathbb R$ $\displaystyle \sin\left(a+b\right) = \sin a \cos b+\sin b\cos a$ and $\displaystyle \sin\left(a-b\right) = \sin a \cos b-\sin b\cos a$ so
$\displaystyle \sin \left(a+b\right)-\sin \left(a-b\right) = 2\sin b\cos a$.
We apply this result for $\displaystyle a=5x$ and $\displaystyle b=7x$:
$\displaystyle I:=\int \cos\left(5x\right)\sin\left(7x\right)\, dx =\int \frac 12\left(\sin \left(12 x\right)-\sin\left(-2x\right)\right) \, dx$
$\displaystyle I= \frac 12 \int \sin \left(12x\right)\, dx +\frac 12 \int \sin \left(2x\right)\,dx= -\frac 1{24} \cos\left(12x\right)-\frac 14\cos\left(2x\right)$

7. $\displaystyle I:=\int \frac{e^x-3}{e^x+2}\,dx= \int\left(1-\frac 5{e^x+2}\right)\, dx =x-5\int \frac{e^{-x}}{1+2e^{-x}}\,dx$ so
$\displaystyle I= x-\frac 52 \int\frac{2e^{-x}}{1+2e^{-x}}\, dx$ and finally $\displaystyle I = x+\frac 52 \ln\left(1+2e^{-x}\right)$
• Jan 26th 2010, 02:56 AM
mr fantastic
Quote:

Originally Posted by egt2010
Let S be the integration symbol

1. S (dx)/ (x. sec ^3 (ln (x)) dx
2. S (x^3)/(sqrt (x^2 -4 )) dx
3. S (2sec x)/(sin x + cos x) dx
4. S sqrt (sqrt (x) -1) dx
5. S sec^3 (x). tan^3 (x) dx
6. S sin (5x)*cos (7x) dx
7. S (e^(x) -3)/(e^(x) +2) dx
8. S (sec^4 (x))/(tan^2 (x)) dx
9. S (sin 2x)/(1 + 3sin^2 (x))^3 dx

Sorry for the many questions, but I wanna make sure of everything related to Integration before my Exam the next Thursday.