TOPIC: Integral Calculus- Transformation of Trigonometric Functions

[leynylaine]

can someone help me in these items? thank you

 

[nasi112]

For the first [MATH]1[/MATH]
What about choosing

[MATH]u = 2\sin 5z + 3[/MATH]
Try this substitution, and let us see where you reach

 

[Dr.Peterson]

can someone help me in these items? thank youView attachment 27577

Please show us an attempt at each of these, so we can see whether you are making reasonable choices and just making a mistake in the work. Also, what techniques have you learned?

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[lex]

As noted before, you need to know certain techniques.
There are obvious substitutions to make e.g. in 1 and 3.
Also you will need to know various reduction formulae e.g. [MATH]\int \text{cosec}^n x \hspace1ex \text{ and } \int \sec^n x \hspace1ex \text{ and } \int \cot^n x[/MATH]and for qun 4 the product-sum formula for [MATH]\sin\alpha \cos\beta[/MATH]

 

[HallsofIvy]

For the first one, using the substitution nasi112 suggested, it is helpful to realize that \(\displaystyle cos^3(5z)dz= (cos^2(5z))(cos(5z) dz)= (1- sin^2(5z))(cos(5z)dz)\).

For the second, \(\displaystyle \frac{d\theta}{sin^7(5\theta)}= \frac{sin(5\theta)d\theta}{sin^8(5\theta)}= \frac{sin(5\theta)d\theta}{(1- cos^2(5\theta))^4}\)