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problems on applications of integration like area ,volume etc

integral of sqrt(tanx)     -----> (integral of sqrt(tanx))

integral of coshx cosx       --------> (integral of cosh x cosx)

integral of {(2x+1) / (4+x.²)} --------->(integral of {(2x+1) / (4+x.²)})

integral of (1+x) / x (x-2)^0.5       --------> (integral of (1+x) / x (x-2)^0.5)

integral of ln(sin x) from 0 to pi/2  ----------> (integral of ln(sin x) from 0 to pi/2)

integral of (e^(ax))cosbx using integration by parts  ----------> (integration by parts)

 .∫ (x²) sin x  dx using integration by parts  ----------> (integration by parts)

integral of { sqrt{1+x^2}  } / {x^2} using integration by parts  ----------> (integration by parts)

∫ arc(tan4x) dx  ----------> (integration by parts)

∫  dx / [ (x-2) (1 + x²)] -----------integration by partial fraction

integral of (x² +1) / [(x+3)(x-1)²]--------------integration by partial fraction

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∫ (sin3x)^4 dx  ----------> (integration by trignometric manipulation)

∫  { cosx/ [ sinx +cosx]  } dx----------------integration by manipulation

∫  { x^2 / (x+1)  } dx--------------integration by long division or manipulation

.∫ (x^4)(ln x)² dx ----------> (integration by parts)

∫ sec³x dx  ----------> (integration)

integral of [ 1 / (x² -x+1)]---------------completion of squares

integral of 1 / sqrt[4 + 3x - x² ]---------------completion of squares

∫ (4x - 7) / (x² + x +1) dx -----------completion of squares after manipulation

integral of sqrt[(1+x)/x ]-----------completion of squares after manipulation

definite integral ∫ x³ e^(-x)dx with limits 0 to infinity --------->(integral)

evaluate integral of { e^(1 /x^2)} / {x^3} -------------(integration by substitution)

integral of ( ( 4 - x^2 )^( 3/2) ) with limits 0 to 1--------------integration by substitution

integral of [cosx (1+sin²x)] with 0 to π /6 as limits-------------integration by substitution

integral of integral of  [e^(arctanx)] / [1 + x²]²-------------integration by substitution

integral of x*sqrt(x+2) ---------------------integration by substitution

evaluate the integral of [e^(-x) +1] ^2 ------------- integration after expansion using identity

  integral of e^{x^(1/3)}--------------  e^{x^(1/3)}

integral of  x / (x+1)^2-------------integration by adjustment

integrate 1/ (x^7 - x) -----------integration by adjusting

integral of 1 / [ (x² +2)²] using trigonometric substitution--------integration using trigonometric substitution

 

(m+n)I(m,n) = m I(m -1,n-1) + (cosx)^m*sin (nx)
where I(m,n ) = ∫ (cosx)^m * cos (nx) dx-----------integral using reduction formulae

multiple integral ∫∫ x² y dy dx over the region bounded by y=x²  and y=1 multiple integral

Using green's theorem show that the area bounded by a simple closed curve C
is given by (1/2).∫ (xdy - y dx)  where the integral is taken in the anticlockwise direction around C answer