Advertisements
Advertisements
प्रश्न
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Advertisements
उत्तर
`int (sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
= `int[((sin^2θ + cos^2θ)^3 - 3sin^2θ*cos^2θ(sin^2θ + cos^2θ))/(sin^2θ*cos^2θ)]*dθ` ...[∵ a3 + b3 = (a + b)3 – 3ab(a + b)]
= `int[((1)^3 - 3sin^2θ*cos^2θ(1))/(sin^2θ*cos^2θ)]*dθ`
= `int[(1)/(sin^2θ*cos^2θ) - 3]*dθ`
= `int [(sin^2θ + cos^2θ)/(sin^2θ*cos^2θ) - 3]*dθ`
= `int (1/cos^2θ + 1/sin^2θ - 3)*dθ`
= `int (sec^2θ + "cosec"^2θ - 3)*dθ`
= `int sec^2θ*dθ + int "cosec"^2θ*dθ - 3int1*dθ`
= tan θ – cot θ - 3θ + c.
APPEARS IN
संबंधित प्रश्न
Integrate the function in x log x.
Integrate the function in x log 2x.
Integrate the function in ex (sinx + cosx).
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Evaluate the following.
`int x^2 *e^(3x)`dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/("9x"^2 - 25)`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int(x + 1/x)^3 dx` = ______.
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
`int 1/sqrt(x^2 - 9) dx` = ______.
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
Solution of the equation `xdy/dx=y log y` is ______
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`inte^x sinx dx`
Evaluate:
`int (logx)^2 dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
`∫ sin^(−1)` xdx is equal to ______.
Which function is an example under priority \(I\) in the LIATE rule?
Which pair is listed as Exponential in the LIATE rule?
Evaluate \[\int e^x\sin x\,dx.\]
