Rajasthan Boardहिन्दी माध्यमकक्षा 12 साइन्सगणितसमाकलन2 Marks
Question
$\int \frac{\left(x^{4}-x\right)^{\frac{1}{4}}}{x^{5}} d x$ ज्ञात कीजिए।
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Answer
हम प्राप्त करते हैं कि $\int \frac{\left(x^{4}-x\right)^{\frac{1}{4}}}{x^{5}} d x =\int \frac{\left(1-\frac{1}{x^{3}}\right)^{\frac{1}{4}}}{x^{4}} d x$
अब $1-\frac{1}{x^{3}} = 1 − x^{−3} = t,$ रखने पर $\frac{3}{x^{4}} d x=d t$
इसलिए $\int \frac{\left(x^{4}-x\right)^{\frac{1}{4}}}{x^{5}} d x =\frac{1}{3} \int t^{\frac{1}{4}} d t$
$=\frac{1}{3} \times \frac{4}{5} t^{\frac{5}{4}}+\mathrm{C} =\frac{4}{15}\left(1-\frac{1}{x^{3}}\right)^{\frac{5}{4}}+\mathrm{C}$
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