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प्रश्न

\[\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx\]

 

योग
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उत्तर

\[\text{Let }I = \int_0^1 \left( x e^x + \cos \frac{\pi x}{4} \right) d x . Then, \]

\[I = \int_0^1 x e^x dx + \int_0^1 \cos\frac{\pi x}{4} dx\]

\[\text{Integrating first term by parts}\]

\[I = \left\{ \left[ x e^x \right]_0^1 - \int_0^1 1 e^x dx \right\} + \left[ \frac{\sin \frac{\pi x}{4}}{\frac{\pi}{4}} \right]_0^1 \]

\[ \Rightarrow I = \left[ x e^x \right]_0^1 - \left[ e^x \right]_0^1 + \left[ \frac{\sin \frac{\pi x}{4}}{\frac{\pi}{4}} \right]_0^1 \]

\[ \Rightarrow I = e - e + 1 + \frac{4}{\pi} \sin \frac{\pi}{4}\]

\[ \Rightarrow I = 1 + \frac{4}{\pi\sqrt{2}}\]

\[ \Rightarrow I = 1 + \frac{2\sqrt{2}}{\pi}\]

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Notes

The answer given in the book has some error. The solution here is created according to the question given in the book.

  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 20: Definite Integrals - Exercise 20.1 [पृष्ठ १७]

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आरडी शर्मा Mathematics [English] Class 12
अध्याय 20 Definite Integrals
Exercise 20.1 | Q 49 | पृष्ठ १७

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