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int2/((e^x+e^(-x))^2)\ dx ( e^(-x))/(e^...

`int2/((e^x+e^(-x))^2)\ dx` `( e^(-x))/(e^x+e^(-x))+C` (b) `-1/(e^x+e^(-x))+C` (c) `(-1)/((e^x+1)^2)+C` (d) `1/(e^x-e^(-x))+C`

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Given,`int2/((e^x+e^(-x))^2)\ dx`
=`int(2e^x)/(e^x(e^x+e^(-x))^2)\ dx`
Substitute `e^x=t⇒e^xdx=dt`
`I=∫2/(t^2+1)^2t​dt=-1/(1+t^2)​+C`
=`-(e^(-x))/(e^x+e^(−x))​+C`
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RD SHARMA-INDEFINITE INTEGRALS-Solved Examples And Exercises
  1. int(sinx)/(3+4cos^2x)\ dx log(3+4cos^2x)+C (b) 1/(2sqrt(3))tan^(-1)((...

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  2. int e^x ((1-sinx)/(1-cosx)) dx= (A) e^xtan(x/2)+C (B) e^xcot(x/...

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  3. int2/((e^x+e^(-x))^2)\ dx ( e^(-x))/(e^x+e^(-x))+C (b) -1/(e^x+e^(-x)...

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  4. int(e^x(1+x))/(cos^2(x e^x))\ dx= 2(log)ecos(x e^x)+C (b) sec(x e^x)+...

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  5. int(sin^2x)/(cos^4x)\ dx= 1/3tan^2x+C (b) 1/2tan^2x+C (c) 1/3tan^3x+C...

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  6. The primitive of the function f(x)=(1-1/(x^2))a^(x+1/x)\ ,\ a >0 is...

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  7. The value of int1/(x+xlogx)\ dx is 1+logx (b) x+logx (c) xlog(1+logx)...

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  8. intsqrt(x/(1-x))\ dx is equal to sin^(-1)sqrt(x)+C (b) sin^(-1){sqrt(x...

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  9. Evaluate: inte^x(f(x)+f^(prime)(x))dx=e^xf(x)+C

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  10. The value of int(sinx+cosx)/(sqrt(1-sin2x))\ dx is equal to sqrt(sin2x...

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  11. If intxsinx\ dx=-xcosx+alpha, then alpha is equal to sinx+C (b) cosx+C...

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  12. int(cos2x-1)/(cos2x+1)\ dx= tanx-x+C (b) x+tanx+C (c) x-tanx+C (d) ...

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  13. int(cos2x-cos2theta)/(cosx-costheta)\ dx is equal to 2\ (sinx+xcosthet...

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  14. int(x^9dx)/((4x^2+1)^9)i se q u a lto (a)1/(5x)(4+1/(x^2))^(-5)+c ...

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  15. int(x^3)/(sqrt(1+x^2))\ dx=a(1+x^2)^(3setminus2)+bsqrt(1+x^2)+C , then...

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  16. int(x^3)/(x+1)\ dx is equal to x+(x^2)/2+(x^3)/3-log|1-x|+C (b) x+(x^...

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  17. Ifint(dx)/((x+2)(x^2+1))=aln(1+x^2)+btan^(-1)x+1/5ln|x+2|+C Then a=1/...

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  18. Evaluate: int1/(sqrt(x)+sqrt(x+1))\ dx

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  19. Evaluate: int(1-x^4)/(1-x)\ dx

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  20. Evaluate: int(x+2)/((x+1)^3)\ dx

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