Antiderivative of cos^2(x)sin(x)

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Antiderivative of cos^2(x)sin(x)

Maro M. asked ? 03/20/13 Find the antiderivative for cosine sqaured x multiplied by sin to the cubed x (cosx)^2 times (sinx)^3 2 Answers By Expert Tutors Tamara J. answered ? 03/21/13 Math Tutoring - Algebra and Calculus (all levels) cos2x?sin3x dx = cos2x?sinx?sin2x dx Recall: sin2x + cos2x = 1 ==> sin2x = 1 - cos2x

cos2x?sinx?(1 - cos2x) dx = (cos2x?sinx - cos4x?sinx) dx = cos2x?sinx dx - cos4x?sinx dx Using u-substitution, let: u = cosx , thus, du = -sinx dx ==> cos2x?sinx dx = -1?cos2x?(-1?sinx dx) = -(u)2?(du) = -u2 du

= -(u)3/3 = -(cosx)3/3 = -cos3x/3 ==> cos4x?sinx dx = -1?cos4x?(-1?sinx dx) = -(u)4?(du) =

-u4 du

= -(u)5/5 = -(cosx)5/5 = -cos5x/5 cos2x?sinx dx - cos4x?sinx dx = (-cos3x/3) - (-cos5x/5)

= -cos3x/3 + cos5x/5 = cos5x/5 - cos3x/3 John M. answered ? 03/20/13 Analytical assistance -- Writing, Math, and more Maro. This integral requires a trig substitution and a u-substitution. Its the trig

substitution that is difficult. When the sine is to an odd power, you want to rewrite the sin term as an even power of sin x times sin x, here, (sin^2 x) (sin x). Then substitute 1-cos^2 x for sin^2 of x and then distribute. In this case you now need to integrate this: [cos^2 x][1-cos^2 x][sin x]. Distribute this out and you will get [cos^2 x][sin x] - [cos^4

x][sin x], with this you can use sum-rule, u-substitution and the power rule to solve each term, which is fairly straightforward. Feel free to email or ask a follow-up question if this does not make sense. John In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation. Gerd

Altmann/Pixabay If you're trying to figure out what x squared plus x squared equals, you may wonder why there are letters in a math problem. That's because, in the case of an equation like this, x can be whatever you want it to be. To find out what x squared plus x squared equals, you have to multiply x times itself. Then you add that number to itself

to get your final answer.Examples of X Squared Plus X Squared Here are some examples of that equation to make it easier to understand. If x equals 2, then x squared, or x times itself, equals 4. Add four to itself, and you get 8. Therefore, 2 squared plus 2 squared equals 8. To use another example, let's see what happens when x equals 3. In that case,

x squared equals 9. Then, 9 plus 9 equals 18. The beauty of this equation is that x can equal anything, and you can solve it using whatever value you want for x. Math that Uses Letters We call mathematics that uses letters to take the place of different values algebra. Algebra uses symbols in most cases, letters to represent quantities that don't

necessarily have the same value all the time. These quantities are called variables, and you can figure out what those variables mean when you use algebra. Equations are like sentences that explain the relationships between numbers and variables. You figure out what the variables in an equation are by solving it. When you solve an equation in

algebra, you break it down to its simplest form and discover what the variables mean. A Brief History of Algebra Since ancient times, mathematicians have worked with unknown variables in different ways. Islamic scholars began to give the science of working with variables a name. They called this type of math the "science of restoration and

balancing," and the Arabic word for "restoration," or "al-jabru," became the root word for the word "algebra." As mathematicians in the Middle Ages experimented with the principles of algebra, they realized they could solve equations for two- and three-dimensional items, which led to even more discoveries of what algebra could do. Modern scholars

have found even more complex equations that algebra can solve. Algebra in Everyday Life You may have heard people say that you'll never use algebra in your everyday life, but you'd be surprised at how often you use algebra. Algebra comes in handy when you're trying to figure out how much a group of items costs per item. When you're trying to

figure out how to split a restaurant bill or how much gas you can buy for a certain amount. You can use algebra to figure out the dimensions of a room or even as you make up your shopping list. Algebra is a versatile form of math that you use more often than you might think and, sometimes, you don't even realize that you're solving math problems.

Why It's Important to Learn Algebra Learning algebra is important for more than just solving equations. Educators consider algebra the gateway to higher forms of math, so if you or your child wants to explore a career in science or technology, algebra can unlock so many more new ideas. Algebra can also help students with critical thinking and logic

skills. Using algebra is like exercise that helps make your brain stronger. Putting algebra to use in your everyday life can help you in so many ways. MORE FROM Get the most by viewing this topic in your current grade. Pick your course now. Derivative VS. Antiderivative For the function f(x)=x3f(x)=x^3f(x)=x3, i) find the

derivative. ii) find the antiderivative. Find the antiderivative of the function: f(x)=xn,n-1f(x)=x^n, neq-1 f(x)=xn,n=-1Find the antiderivative of each of the following functions. f(x)=x100f(x)=x^{100}f(x)=x100 f(x)=7x-2f(x)=7x^{-2}f(x)=7x-2 Find the derivatives and antiderivatives of the following common functions. Find the antiderivative of

each of the following functions. f(x)=4x5-6x2+7x-xx2+5cosx-100ex f(x)= \frac{4x^5-6x^2+7x-\sqrt{x}}{x^2}+5cos x-100e^x f(x)=x24x5-6x2+7x-x+5cosx-100ex g(x)=5x+31-x2 g(x)=5^x+\frac{3}{\sqrt{1-x^2}} g(x)=5x+1-x23 Yusuke Kawasaki/Flickr Xacuti, xiaolongbao, ximenia, xoconostle and xpinec are just some of the foods that begin

with the letter "X." Because so few words begin with the letter "X" in English, all of these foods come from countries outside the United States. Xacuti This brown curry comes from the village of Arambol in the Indian region of Goa and is usually made with chicken or seafood. Xacuti is often served with rice, bread or even over an omelet. It can vary in

spiciness. Ingredients include pepper, onion, white poppy seeds, fresh and dried chilies, turmeric, cinnamon, cloves, nutmeg and other spices. Xiao Long Bao A small (xiao) basket (long) bun (bao) is a steamed dumpling filled with broth and pork. Originally from Shanghai, China, xiao long baos can now sometimes be found in American Chinese

restaurants in big cities like Chicago. Eating these dumplings can be difficult for people who haven't tried them before. They are served in a bamboo steamer while extremely hot, making it risky to eat one too soon. However, the longer you wait for the xiao long bao to cool, the more likely it is that the bottom will tear, losing the delicious broth. To

overcome these challenges, pick the dumpling up with chopsticks and place it on a soup spoon. (You may wish to add vinegar and ginger to the spoon beforehand for added flavor.) As the dumpling cools on the spoon, consider piercing the skin of the dumpling with a fork or your teeth to help it reach a safe temperature faster. Once a few minutes have

gone by and the dumpling is ready, slide the dumpling (and any vinegar and ginger) into your mouth and enjoy. Ximenia Ximenia is the name of both a tree and its fruit that grows in countries such as Ethiopia, Tanzania and South Africa. Its English name comes from Francisco Ximenez, a Spanish monk. The fruit is orange or red with white spots and

only slightly longer than an inch. The taste is bitter and tart. The skin should be peeled and discarded before eating, although the nut is edible. Ximenia fruit are used for jams, jellies, deserts and as a sweetener for porridge. They are also eaten raw. Additionally, the roots and leaves of the tree can be used for medicinal purposes, such as treating

fevers or inflamed eyes. Xnipec The name of this fiery salsa comes from the Mayan words for "dog's nose" or "dog's snout," probably because the heat of it will make your nose run wet like a dog's. Xnipec is originally from the Yucatan Peninsula, the southeastern part of Mexico where Mayans still live today, but it has since made its way north of the

border. The spiciness of this salsa comes from habanero peppers, so be careful when preparing and eating xnipec. Xonocostle Like Ximenia, Xonocostle is a fruit. It comes from a cactus that goes by the same name and grows in Central Mexico. It's used in marinades, salsas, mole de olla and even beverages. Sometimes it's also dried or candied before

being eaten. The cactus itself is pale green, while the fruit is deep red and grows at the end of paddle-shaped growths. The taste of xonocostle is sour and acidic. MORE FROM

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