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  • Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Multiple Choice Questions Question 5 Maths Textbook Solution.

Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Multiple Choice Questions Question 5 Maths Textbook Solution.

Answers (1)

Answer:

x^{\sin x}+C

Given:

\int x^{\sin x}\left(\frac{\sin x}{x}+\cos x \cdot \log x\right) d x

Hint:

You must know about the derivation of x^{\sin x}

Explanation:

Let \mathrm{I}=\int x^{\sin x}\left(\frac{\sin x}{x}+\cos x \cdot \log x\right)

        =\int t \cdot \frac{d t}{t}                                                                                     \text { [ Put } x^{\sin x}=t, \text { Taking log of both sides }

       =\int 1 d t                                                                                          \sin x \cdot \log x=\log t

       =t+C                                                                                           Diff.  w. r. t.  t,

         =x^{\sin x}+C                                                                           \left.\left(\log x \cos x+\frac{\sin x}{x}\right) \frac{d x}{d t}=\frac{1}{t} \Rightarrow\left(\log x \cos x+\frac{\sin x}{x}\right) d x=\frac{d t}{t}\right]                  

 

 

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