Find the integral of sinx/x.

We can make use of taylor series of expansion.

Answer : The expression on integrating sin x / x is  x - (x3/3×3!) + (x5/5×5!) - (x7/7×7! ) + ....+ C

Let's look into the stepwise solution

Explanation:

Let us see the taylor series of expansion for sin x.

We have,

sin x = x - (x3/3!) + (x5/5!) - (x7/7!) + ..... upto infinite terms. [where ! sign denotes the factorial of a number]

sin x / x = 1/x { x - (x3/3!) + (x5/5!) - (x7/7!) + .....}

          =  1 - (x2/3!) + (x4/5!) - (x6/7!) + ....

Now,

∫sin x / x =  ∫(1 - (x2/3!) + (x4/5!) - (x6/7!) + ....)dx

We know that, ∫xndx = x(n+1)/n+1 + C

where, C is the integration constant

Thus,

∫sin x / x = x - (x3/3×3!) + (x5/5×5!) - (x7/7×7! ) + ....+ C

Hence, the expression on integrating sin x/ x is x - (x3/3×3!) + (x5/5×5!) - (x7/7×7! ) + ....+ C

 

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