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Evaluate the integral.
$ \displaystyle \int \sin x \sec^5 x dx $
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Calculus 2 / BC
Techniques of Integration
Oregon State University
University of Michigan - Ann Arbor
University of Nottingham
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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.
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this problem is from Chapter seven section to problem number forty four in the book Calculus Early. Transcendental. Lt's a condition by James Door. We have an integral of sign. Next time, see cancer the fifth power here weaken. Use the fact that she can't is one over co sign. So let's pull out One factor of C can right is one over co sign and then we have our remaining seeking to the fourth power left over. So here we have integral of tangent time seeking to the fourth power of X. Next we can go ahead. And since we already have one factor of tangent here, let let's pull out another seeking from the Sikh into the fourth. So we have tangent times. He can't times he can. Cute. This is just we do a u substitution. Let's take you to B. C. Can't then do you is seeking time, Stan, which is precisely in our incident we have here. We have tangent times. He can't gx so using our new substitution, we can rewrite this as you cube coming from C Can cube. Do you? And then we could use the power rule to evaluate this. You to the fourth power over four. Plus he and finally, we should go back to our U substitution to replace you with Seek Innovex. So we obtain. She came to the fore Power Vex over four. Plus he and that's your answer.
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