125. Wallis products Complete the following steps to prove a w...

Question

125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.

a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

Accepted Answer

Hello. In this video, we want to suppose that we have a function case of P equal to a definite integral from 0 toi of sine of the power of PX for integers P greater than or equal to 2. We want to figure out which reduction formula correctly relates case P and K P minus 2. Now, the function case of P is an integral of a sine function, and we have a reduction formula for the sine function. That reduction formula is the following. If we have an integral of sin to the power of PX, then this is going to simplify to be negative, 1 divided by P, multiplied by sin to the power of P minus 1 of X, multiplied by cosine of X. Minus I'm sorry, plus the quantity P + 1 divided by P multiplied by the integral of sine of P minus 2 of XDX, and this is for all integers of P greater than or equal to 2. Now, let's go ahead and take a look again at our KP function. K P is given to us as the integral from 0 toi of sine of PXDX. Now, the thing about this integral is that we are integrating the sine function only over the boundary from 0 to pi. On this boundary from 0 to pi, we know that the sine function is positive, but more importantly, the sine function is symmetric. And because we have a positive symmetric sine function on the boundaries from 0 to pi, we can rewrite our KP function to be case of P is equal to 2 multiplied by the integral of 0 to pi divided by 2. Signed to the power of P X D X. Now, this form of the sine function has its own reduction formula. The reduction formula is the following. The integral from 0 to pi divided by 2 of sine to the PXDX is equal to the quantity P minus 1 divided by P, multiplied by the integral from 0 to pi divided by 2 of sign of P minus 2 of XD X. So if we were to replace The right hand side with K minus 2, because this integral is in the form of K P minus 2. That we can rewrite our KP statement as KP. is equal to 2 multiplied by P minus 1 divided by P multiplied by the integral from 0 to pi divided by 2 of sine of P minus 2 of XDX. Now, notice that this integral here. is in the form of K P minus 2. And so what that means is that if we were to reorder these two terms and put the 2 in front of the coefficient of the integral, then we can have the quantity P minus 1 divided by P multiplied by 2, multiplied by 0 to divided by 2. Of sign of P minus 2 of XDX, and this is going to simplify to give us K P is equal to the quantity P minus 1 divided by P multiplied by K P minus 2, and this is going to be the solution to the problem. And overall, this relates the two functions, and the solution is going to be C. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

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