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Evaluate the integral.
$ \displaystyle \int \sin^5 t \cos^4 t\ dt $
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Calculus 2 / BC
Techniques of Integration
Strategy for Integration
Harvey Mudd College
Idaho State University
In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative.
The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis.
The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.
In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.
Evaluate the integral.…
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let's evaluate this trick in a metric and roll by rewriting it and then using the use of So here I see that I have an odd power on the sign. So let me pull out all the factors of sign except want just one power. I need one power hanging around at the end. It's all right out here and that will come in when I using substitution. So now it looks like I can do it. You? Not yet, but it looks like we're gearing toward U equals ko society. So if I'd like to use that substitution, I should rewrite this original signed the fourth in terms of co sign. So we have sign fourth that sign square, but that and also square that whole thing. And then using the pathetic and identity you can write, this is one minus close sites where and so we can use this to rewrite sign of the fourth And then we still have co sign of the fourth year. And now we're ready to use the U substitution. So let you because I'm and then here because of the negative sign. Let's write. This is negative to you, equal scientist DT, and then we have negative one minus you square that's also squared. And then we have you to the fourth Power, do you? So let's just go ahead and multiply this zone and then we have negative in a girl. You're the fourth and then also at the end, we have you the eighth. Now let's just go ahead and use the power rule three times and then lets his push that minus sign on through minus you to the nine over nine. Plus he and then finally, the final answer will just come back to our use up and then replace you in terms of tea. So that's co signed the fifth power over five hours to co sign to the seventh power over seven and then minus co signs to the ninth Power over nine. Plus our constancy. And that's your final answer.
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