NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 7 - Integrals [Latest edition]

Integrate `((2"a")/sqrt(x) - "b"/x^2 + 3"c"root(3)(x^2))` w.r.t. x

Evaluate `int (3"a"x)/("b"^2 + "c"^2x^2) "d"x`

Verify the following using the concept of integration as an antiderivative

`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`

Evaluate `int sqrt((1 + x)/(1 - x)) "d"x`, x ≠1

Evaluate `int "dx"/sqrt((x - alpha)(beta - x)), beta > alpha`

Evaluate `int tan^8 x sec^4 x"d"x`

Find `int x^2/(x^4 + 3x^2 + 2) "d"x`

Find `int "dx"/(2sin^2x + 5cos^2x)`

Evaluate `int_(-1)^2 (7x - 5)"d"x` as a limit of sums

Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`

Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`

Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`

Find `int x^2tan^-1x"d"x`

Find `int sqrt(10 - 4x + 4x^2)  "d"x`

Evaluate `int (x^2"d"x)/(x^4 + x^2 - 2)`

Evaluate `int (x^2 + x)/(x^4 - 9) "d"x`

Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`

Find `int_0^1 x(tan^-1x)  "d"x`

Evaluate `int_(-1)^2 "f"(x)  "d"x`, where f(x) = |x + 1| + |x| + |x – 1|

`int "e"^x (cosx - sinx)"d"x` is equal to ______.

`int "dx"/(sin^2x cos^2x)` is equal to ______.

If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4e^x + 5e –x| + C, then ______.

`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.

If f and g are continuous functions in [0, 1] satisfying f(x) = f(a – x) and g(x) + g(a – x) = a, then `int_0^"a" "f"(x) * "g"(x)"d"x` is equal to ______.

If x = `int_0^y "dt"/sqrt(1 + 9"t"^2)` and `("d"^2y)/("d"x^2)` = ay, then a equal to ______.

`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.

If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.

`int_(-2)^2 |x cos pix| "d"x` is equal to ______.

`int (sin^6x)/(cos^8x) "d"x` = ______.

`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.

`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.

`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.

Verify the following:

`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`

Verify the following:

`int (2x + 3)/(x^2 + 3x) "d"x = log|x^2 + 3x| + "C"`

Evaluate the following:

`int ((x^2 + 2))/(x + 1) "d"x`

Evaluate the following:

`int ("e"^(6logx) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x`

Evaluate the following:

`int ((1 + cosx))/(x + sinx) "d"x`

Evaluate the following:

`int ("d"x)/(1 + cos x)`

Evaluate the following:

`int tan^2x sec^4 x"d"x`

Evaluate the following:

`int (sinx + cosx)/sqrt(1 + sin 2x) "d"x`

Evaluate the following:

`int sqrt(1 + sinx)"d"x`

Evaluate the following:

`int x/(sqrt(x) + 1) "d"x`  (Hint: Put  `sqrt(x)` = z)

Evaluate the following:

`int sqrt(("a" + x)/("a" - x)) "d"x`

Evaluate the following:

`int x^(1/2)/(1 + x^(3/4)) "d"x`   (Hint: Put `sqrt(x)` = z⁴)

Evaluate the following:

`int sqrt(1 + x^2)/x^4 "d"x`

Evaluate the following:

`int ("d"x)/sqrt(16 - 9x^2)`

Evaluate the following:

`int "dt"/sqrt(3"t" - 2"t"^2)`

Evaluate the following:

`int (3x - 1)/sqrt(x^2 + 9) "d"x`

Evaluate the following:

`int sqrt(5 - 2x + x^2) "d"x`

Evaluate the following:

`int x/(x^4 - 1) "d"x`

Evaluate the following:

`int x^2/(1 - x^4) "d"x` put x² = t

Evaluate the following:

`int sqrt(2"a"x - x^2)  "d"x`

Evaluate the following:

`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`

Evaluate the following:

`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`

Evaluate the following:

`int (sin^6x + cos^6x)/(sin^2x cos^2x) "d"x`

Evaluate the following:

`int sqrt(x)/(sqrt("a"^3 - x^3)) "d"x`

Evaluate the following:

`int (cosx - cos2x)/(1 - cosx) "d"x`

Evaluate the following:

`int ("d"x)/(xsqrt(x^4 - 1))`  (Hint: Put x² = sec θ)

Evaluate the following as limit of sum:

`int _0^2 (x^2 + 3) "d"x`

Evaluate the following as limit of sum:

`int_0^2 "e"^x "d"x`

Evaluate the following:

`int_0^2 ("d"x)/("e"^x + "e"^-x)`

Evaluate the following:

`int_0^(pi/2) (tan x)/(1 + "m"^2 tan^2x) "d"x`

Evaluate the following:

`int_1^2 ("d"x)/sqrt((x - 1)(2 - x))`

Evaluate the following:

`int_0^1 (x"d"x)/sqrt(1 + x^2)`

Evaluate the following:

`int_0^pi x sin x cos^2x "d"x`

Evaluate the following:

`int_0^(1/2) ("d"x)/((1 + x^2)sqrt(1 - x^2))`  (Hint: Let x = sin θ)

Evaluate the following:

`int (x^2"d"x)/(x^4 - x^2 - 12)`

Evaluate the following:

`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`

Evaluate the following:

`int_"0"^pi  (x"d"x)/(1 + sin x)`

Evaluate the following:

`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`

Evaluate the following:

`int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`

Evaluate the following:

`int sin^-1 sqrt(x/("a" + x)) "d"x`  (Hint: Put x = a tan²θ)

Evaluate the following:

`int_(pi/3)^(pi/2) sqrt(1 + cosx)/(1 - cos x)^(5/2)  "d"x`

Evaluate the following:

`int "e"^(-3x) cos^3x  "d"x`

Evaluate the following:

`int sqrt(tanx)  "d"x`  (Hint: Put tanx = t²)

Evaluate the following:

`int_0^(pi/2)  "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos⁴x)

Evaluate the following:

`int_0^1 x log(1 + 2x)  "d"x`

Evaluate the following:

`int_0^pi x log sin x "d"x`

Evaluate the following:

`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`

`int (cos2x - cos 2theta)/(cosx - costheta) "d"x` is equal to ______.

`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.

`int tan^-1 sqrt(x)  "d"x` is equal to ______.

`int "e"^x ((1 - x)/(1 + x^2))^2  "d"x` is equal to ______.

`int x^9/(4x^2 + 1)^6  "d"x` is equal to ______.

If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.

`int x^3/(x + 1)` is equal to ______.

`int (x + sinx)/(1 + cosx) "d"x` is equal to ______.

If `intx^3/sqrt(1 + x^2) "d"x = "a"(1 + x^2)^(3/2) + "b"sqrt(1 + x^2) + "C"`, then ______.

`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.

`int_0^(pi/2) sqrt(1 - sin2x)  "d"x` is equal to ______.

`int_0^(pi/2)  cos x "e"^(sinx)  "d"x` is equal to ______.

`int (x + 3)/(x + 4)^2 "e"^x  "d"x` = ______.

If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.

`int sinx/(3 + 4cos^2x) "d"x` = ______.

The value of `int_(-pi)^pi sin^3x cos^2x  "d"x` is ______.