Solve integral of x^100 * sinx dx

Question

Is there any 'clever' way to solve integral of x^100 * sinx dx, I had this on my exam o_o

Comments

Is this x¹⁰⁰sin(x) or x^100sin(x)?

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first one, x100sin(x)

Answer 1

If x¹⁰⁰sin(x) is implied, there doesn't seem to be a "trick" to solve this as an indefinite integral, but there is a method to solve it, because by integrating by parts, the integrand can be progressively reduced so that the power of x decreases until eventually ∫sin(x)dx is reached, which is readily integrable. The final integral involves terms (such as 100 factorial) which are not readily computable.

If a definite integral is required (for some interval of x) some shortcuts would provide an approximation. To see what would be required, consider x=π/2, then sin(x)=1 and x¹⁰⁰sin(x)=(π/2)¹⁰⁰ (also applies when x=-π/2) which is a large number. When 0<x<1, x¹⁰⁰ is positive but less than 1, mostly considerably less; when x=1, x¹⁰⁰sin(x)=0.84 approx. It would be interesting to know the context of the exam question.

Integration delivers the area under a curve, and in this case, the curve has outstanding characteristics, leaping from close to zero to a very large number in a short interval. While sin(x) varies between -1 and 1, x¹⁰⁰ is always positive and is either very small or very large, so the gradient shifts from almost horizontal to almost vertical in the same interval. This behaviour provides a clue to evaluation of the definite integral, including "tricks" to perform this evaluation.

Comments

that was an online exam, everyone got 1 question from integrals, matrices, 2 var functions and geometry, 1 each, bruh, for example, i had to 'calculate' det of a 10x10 matrix [1,1,0,0,0,0,0,0,0,0,0,0; 2,2,2,0,0,0,0,0,0,0; 3,3,3,3,0,0...;...; 10,10,10,10,10,10,10,10,10,10] O.o

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For this integral, i show D I method, wrote down 4-5 multiplications and then ... + C xD and i got 1 out of 3 points for that

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D I = Definite Integral?

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The 10 by 10 determinant you quoted actually has 12 elements on the first row! My guess is that the first row should have been 1,0,0,0,0,0,0,0,0,0. Also, if there were 10 tens in the last row and no zeroes, then the penultimate row would have 9 nines and one zero, ..., third row 3 threes and 7 zeroes, second row 2 twos and 8 zeroes, first row 1 one and 9 zeroes, surely?

If the original integral was a definite integral what were the limits?