प्रश्नावली - 7V
निम्न समाकलों का मान ज्ञात कीजिए-
1. $\int_0^1 \frac{\left(\tan ^{-1} x\right)^2}{1+x^2} d x$
2. $\int_0^1 \frac{x}{1+x^2} d x$
3. $\int_0^2 \sqrt{\frac{2+x}{2-x}} d x$
4. $\int_1^3 \frac{\log _e x}{x} d x$
5. $\int_0^{\pi / 2} \sin 2 x \tan ^{-1}(\sin x) d x$
6. $\int_0^1 x \log _e\left(1+\frac{x}{2}\right) d x$
7. $\int_1^e \frac{e^x}{x}\left(1+x \log _e x\right) d x$
8. $\int_0^1 \frac{\sin ^{-1} x}{\sqrt{1-x^2}} d x$
9. $\int_0^1 \frac{x \sin ^{-1} x}{\sqrt{1-x^2}} d x$
10. $\int_0^{\infty} \frac{\sin \left(\tan ^{-1} x\right)}{1+x^2} d x$
11. $\int_0^2 x \sqrt{x+2} d x$
12. $\int_0^1 \frac{x}{\sqrt{1-x^2}} d x$
13. $\int_0^a \sqrt{a^2-x^2} d x$
14. $\int_0^1 x \sqrt{\frac{1-x^2}{1+x^2}} d x$
15. $\int_0^1 \frac{d x}{\sqrt{1+x}-\sqrt{x}}$
16. $\int_0^a x \sqrt{\frac{a^2-x^2}{a^2+x^2}} d x$
17. $\int_0^2 \frac{d x}{x+4-x^2}$
18. $\int_0^{\pi / 2} x \sin x \cos x d x$
19. $\int_0^{\pi / 2} \sin x \sin 2 x d x$
20. $\int_{\pi / 2}^\pi e^x\left(\frac{1-\sin x}{1-\cos x}\right) d x$
21. $\int_0^{\pi / 2} \cos ^2 x \sin ^2 x d x$
22. $\int_0^{\pi / 2} e^x(\sin x+\cos x) d x$
23. $\int_0^{\pi / 2} \frac{\sin x}{1+\cos ^2 x} d x$
24. $\int_0^{\pi / 4} \frac{\sin x \cos x}{\cos ^4 x+\sin ^4 x} d x$
25. $\int_0^{\pi / 2} \frac{d x}{4+3 \cos x}$
26. $\int_0^{\pi / 2} \sqrt{\sin \phi} \cos ^5 \phi d \phi$
27. $\int_0^{\pi / 2} x^2 \sin x d x$
28. $\int_{-1}^1 \frac{d x}{x^2+2 x+5}$
29. $\int_0^{\pi / 2} \frac{d x}{a^2 \cos ^2 x+b^2 \sin ^2 x}$
30. $\int_1^2\left(\frac{1}{x}-\frac{1}{2 x^2}\right) e^{2 x} d x$
31. सिद्ध करो कि $\int_0^{\pi / 2} \frac{d x}{1+2 \cos x}=\frac{1}{\sqrt{3}} \log (2+\sqrt{3})$.
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