# Integrate: ## \int \cos^m(x)\sin^n(x)~dx ##

• Fatima Hasan
In summary: If the m-2 exponent is the correct one (as the formula implies), then the integral is much easier to solve.
Fatima Hasan

## Homework Statement

Derive that:

Correction (from a later post)
The integral on the left side should be ##\int \cos^m(x)\sin^n(x)~dx##

-

## The Attempt at a Solution

Anyone can help me on how to start?

## u = cos^m x ##

## du = m cos^{m-1} x ##

## dv = sin^n x dx ##

## v = \frac{sin^{n+1} x}{n} ##

#### Attachments

• png.png
5.6 KB · Views: 641
Last edited by a moderator:
Hi,
##du = m \cos^{m-1} x## can not be correct; the left hand side is a differential and the right hand side is not. Also, it is clearly wrong for ##m = 1## ...

BvU said:
Hi,
##du = m \cos^{m-1} x## can not be correct; the left hand side is a differential and the right hand side is not. Also, it is clearly wrong for ##m = 1## ...
How to derive ## u = cos^{m} \,\,\, x ## ?

Last edited:
You use the chain rule

BvU said:
You use the chain rule

#### Attachments

• Pediatric_Surgery.png
12.7 KB · Views: 253
BvU
Now that we have settled that: is there a typo in the problem statement ? On the left I see no variable ##n## !?

BvU said:
Now that we have settled that: is there a typo in the problem statement ? On the left I see no variable ##n## !?
yeah , it should be ∫cosmx sinndx

#### Attachments

• png.png
3.4 KB · Views: 227
Doesn't look like the expression in post #1 yet, does it ?
What could be the trick to get e.g. ##m+n## in the denominator ?

How can I ?

Fatima Hasan said:
How can I ?

You cannot. Your are being asked to show something that is false. If you let ##L## be your (corrected) left-hand-side and ##R## the right-hand-side, then trying out some small integer values of ##m## and ##n## will show that the result is wrong. For example:
$$\begin{array}{cl} m=2,n=2: & L = -\frac{1}{4} \cos(x)^3 \sin(x)+ \frac{1}{8} \cos(x) \sin(x)+ \frac{1}{8} x&\\ & R = \frac{1}{4} \cos(x) \sin(x)^3- \frac{1}{8} \cos(x) \sin(x)+\frac{1}{8} x \\ m=5,n=3: & L = - \frac{1}{8} \cos(x)^6 \sin(x)^2- \frac{1}{24} \cos(x)^6 \\ &R = \frac{1}{8} \cos(x)^4 \sin(x)^4-\frac{1}{10} \cos(x)^3 \sin(x)^2- \frac{1}{15} \cos(x)^3 \end{array}$$

You can check these using a computer algebra system; I used Maple, but you can do it (at no cost) using Wolfram Alpha, for example.

Last edited:
Checking some online tables of trig integrals, I found this one that is nearly the same as that in the OP:

$$\int \cos^m(x)\sin^n(x)dx = \frac{\cos^{m-1}(x)\sin^{n+1}(x)}{m + n} + \frac{m - 1}{m + n}\int \cos^{m-2}(x)\sin^n(x)dx$$
The main difference between this integral formula and the one in post #1 is the exponent on the cosine factor in the last integral.
In post 1, the exponent is m - n. In the formulat just above, the exponent is m - 2. As @Ray Vickson points out, the integral as posted isn't correct, so it's very likely that the m-n exponent is a typo.

## 1. What does the notation ## \int \cos^m(x)\sin^n(x)~dx ## mean?

The notation ## \int \cos^m(x)\sin^n(x)~dx ## represents the integral of the product of ## \cos^m(x) ## and ## \sin^n(x) ## with respect to the variable ## x ##. In other words, it is the area under the curve of the function ## \cos^m(x)\sin^n(x) ##.

## 2. How do you solve the integral ## \int \cos^m(x)\sin^n(x)~dx ##?

The solution to this integral depends on the values of ## m ## and ## n ##. For certain combinations of even and odd values, the integral can be solved using trigonometric identities. For other combinations, it may require the use of substitution or integration by parts.

## 3. Can the integral ## \int \cos^m(x)\sin^n(x)~dx ## be evaluated using a calculator?

No, most calculators do not have the capability to solve integrals with trigonometric functions. This type of integral requires knowledge of integration techniques and cannot be solved solely with a calculator.

## 4. What is the purpose of integrating ## \cos^m(x)\sin^n(x)~dx ##?

Integrating ## \cos^m(x)\sin^n(x)~dx ## allows us to find the area under the curve of the given function, which can be useful in various applications such as physics, engineering, and economics. It also helps us understand the behavior of the function and its relationship with its variables.

## 5. Are there any special cases of the integral ## \int \cos^m(x)\sin^n(x)~dx ##?

Yes, there are a few special cases of this integral. When either ## m ## or ## n ## is equal to zero, the integral reduces to a simple form. Additionally, when both ## m ## and ## n ## are odd, the integral can be solved using the double angle formula for cosine. These special cases can make solving the integral easier and faster.

• Calculus and Beyond Homework Help
Replies
3
Views
494
• Calculus and Beyond Homework Help
Replies
15
Views
909
• Calculus and Beyond Homework Help
Replies
1
Views
396
• Calculus and Beyond Homework Help
Replies
11
Views
801
• Calculus and Beyond Homework Help
Replies
1
Views
612
• Calculus and Beyond Homework Help
Replies
7
Views
885
• Calculus and Beyond Homework Help
Replies
22
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
731
• Calculus and Beyond Homework Help
Replies
27
Views
3K
• Calculus and Beyond Homework Help
Replies
2
Views
947