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Evaluate the integral.

$ \displaystyle \int (x - 1) \sin \pi x dx $

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$\int(x-1) \cdot \sin (\pi x) d x=\frac{1-x}{\pi} \cdot \cos (\pi x)+\frac{1}{\pi^{2}} \sin (\pi x)+C$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

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Lectures

01:53

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.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

02:25

Evaluate the integral.…

0:00

04:12

Evaluate the integrals.

01:30

02:39

01:41

Evaluate definite integral…

00:42

Evaluate the definite inte…

02:00

03:21

Oh, hello. So today we're evaluating them to go of experts. One time sign of pi X, the X There really should be a parentheses around the pie ex together because you want to take a sign of pious IX. This is a standard, Ah, integration by parts problems. So you want to find you in a DVD to decompose it into, um I want to find a you that basically disappears after immigrating after differentiating once or twice, or however many times this case, he only wanted differentiate ones. You equals X minus one in this case, and Devi is equal to sign of pi X t x which is you're able to integrate basically so differentiating you, you got d equals DX integrating Devi you get vehicles native one over pi co sign of pi X. Put it all together you get native one of her pi X minus one co sign of pi X minus negative one reply. Integral of coastline of pi X, The X I like to put all my constance outside of the interval because it's easier to work on afterwards. Ah, simplifying you get native one of reply expense one co sign a pi X plus one over pi times one of reply again because you get no another integration from the coastline up Iraq's plus C musical to native wanna rip I express one cosign a pi x plus one over pi squared sine of pi x plus e.

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