Advertisements
Advertisements
Question
Evaluate the following : `int sin θ.log (cos θ).dθ`
Advertisements
Solution
Let I = `int sin θ.log (cos θ).dθ`
= `int log(cosθ).sinθ dθ`
Put cos θ = t
∴ – sin θ dθ = dt
∴ sin θ dθ = – dt
∴ I = `int logt.(- dt)`
= `- int (logt).1dt`
= `-[(log t) int 1dt - int {d/dt (log t) int 1 dt }dt]`
= `-[(logt)t - int1/t.t dt]`
= `- t logt + int 1 dt`
= – t log t + t + c
= – cos θ . log (cos θ) + cos θ + c
= – cos θ [log (cos θ) – 1] + c.
APPEARS IN
RELATED QUESTIONS
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Integrate the function in x tan-1 x.
Integrate the function in e2x sin x.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x^3.logx.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
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 (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int(x + 1/x)^3 dx` = ______.
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
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`
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.
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int 1/sqrt(x^2 - a^2)dx` = ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`intsqrt(1+x) dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
`inte^(xloga).e^x dx` is ______
`int logx dx = x(1+logx)+c`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
Evaluate the following:
`intx^3e^(x^2)dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
Which function is an example under priority \(I\) in the LIATE rule?
Which pair is listed as Exponential in the LIATE rule?
When applying integration by parts to \(\log x\) and inverse trig, what should they be multiplied by?
