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Maths-

General

Easy

Question

If $integral$ x sin x dx = – $x c o s invisible function application x plus alpha comma text then end text alpha$ =

sin x + c
cos x + c
x cos x + c
cos x – sin x + c.

hint Hint:

Using formula $integral space u. v space d x space equals u. integral v space d x space space minus integral left parenthesis fraction numerator d u over denominator d x end fraction. integral v space d x right parenthesis space d x$ where u and v are functions of x . Find the value of $integral$ x sin x dx and compare it with given value to find value of alpha.

The correct answer is: sin x + c

Given , $integral$ x sin x dx let u = x and v = sin x
then $fraction numerator d u over denominator d x end fraction space equals space fraction numerator d over denominator d x end fraction left parenthesis x right parenthesis space equals space 1$ , $integral space v space d x space equals integral space sin x space d x space equals space minus space cos x space$
Using formula $integral space u. v space d x space equals u. integral v space d x space space minus integral left parenthesis fraction numerator d u over denominator d x end fraction. integral v space d x right parenthesis space d x$
Using formula,
$integral space x space sin x space d x space equals x. integral sin x space d x space space minus integral left parenthesis 1. integral sin x space d x right parenthesis space d x equals x. left parenthesis negative space cos space x right parenthesis space space minus integral left parenthesis 1. left parenthesis negative cos space x right parenthesis space d x equals space minus x space cos space x space plus space integral cos space x space d x equals space space minus x space cos space x space plus space sin space x space plus space c$
Comparing - x cos x + sin x + c with – $x c o s invisible function application x plus alpha comma text then end text alpha$ = sin x + c

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