Question 1. Integrate the following integrals with respect to x:
(i) ∫ x4 dx
Solution:
∫ x4 dx = x4+1/(4+1) + Constant
= x5/5 + C
(ii) ∫ x5/4 dx
Solution:
∫ x5/4 dx = x5/4 + 1/(5/4 +1) + Constant
= 4/9 x9/4 + C
(iii) ∫ 1/x5 dx
Solution:
∫ 1/x5 dx = ∫ x-5 dx
= x-5+1/(-5+1) + Constant
= x-4/(-4)+ C
= -1/(4x4) + C
(iv) ∫ 1/x3/2 dx
Solution:
∫ x-3/2 dx = x-3/2 + 1/(-3/2 +1) + Constant
= x-1/2/(-1/2) + C
= -2/(√x)+ C
(v) ∫ 3x dx
Solution:
∫ 3x dx = 3x/log3 + Constant
(vi) ∫ 1/x2/3 dx
Solution:
∫ 1/x2/3 dx = ∫ x-2/3 dx
= x-2/3 + 1/(-2/3+1) + Constant
= x1/3/(1/3) + C
= 3x1/3 + C
(vii) ∫ 32log3 x dx
Solution:
∫ 32log3 x dx =
∫ 3^{log_3 x^2} dx = ∫ x2 dx
= x2+1/(2+1) + Constant
= x3/3 + C
Question 2. Evaluate
(i) ∫\sqrt{\frac{(1 + cos 2x)}{2} }dx
Solution:
∫\sqrt{\frac{(1 + cos 2x)}{2} }dx =
∫\sqrt{\frac{(1 + 2cos^2x - 1)}{2}} dx We know, cos 2x = 2cos2 x - 1
=
∫\sqrt{\frac{(2cos^2x)}{2}} dx = ∫cos x dx
= sin x + Constant
(ii)∫\sqrt{\frac{(1 - cos 2x)}{2} }dx
Solution:
∫\sqrt{\frac{(1 - cos 2x)}{2} }dx =
∫\sqrt{\frac{(1 - 1 + 2sin^2x)}{2}} dx We know, cos 2x = 1 - 2sin2 x
=
∫\sqrt{\frac{(2sin^2 x)}{2}} dx = ∫ sin x dx
= -cos x + Constant
Question 3. Evaluate ∫ \frac{e^{6log_ex}-e^{5log_ex}}{e^{4log_ex}-e^{3log_ex}}
Solution:
∫ \frac{e^{6log_ex}-e^{5log_ex}}{e^{4log_ex}-e^{3log_ex}} dx=
∫\frac{(x^6 - x^5)}{(x^4 - x^3)} dx We know, e loge x = x
=
∫ \frac{x^5(x - 1)}{x^3(x - 1)} dx = ∫ x2 dx
= x2+1/2+1 + Constant
= x3/3 + C
Question 4. Evaluate: ∫ \frac{1}{a^x b^x}dx
Solution:
∫ \frac{1}{a^x b^x}dx = ∫ a-x b-x dx= ∫ (ab)-x dx
= (ab)-x/loge (ab)-1 + Constant
= -a-x b-x/loge (ab) + C
or
= -a-x b-x/ ln(ab) + C
Question 5. Evaluate
(i) ∫ \frac{cos 2x + 2sin^2 x}{sin^2 x} dx
Solution:
∫ \frac{cos 2x + 2sin^2 x}{sin^2 x} dx =
∫ \frac{1 - 2sin^2x + 2sin^2x}{sin^2x} dx We know, cos 2x = 1 - 2sin2 x
= ∫ 1/sin2x dx = ∫ cosec2x dx
= -cot x + Constant
(ii) ∫\frac{2cos^2x - (cos2x)}{cos^2x}dx
Solution:
∫\frac{2cos^2x - (cos2x)}{cos^2x}dx
∫\frac{2cos^2x - (2cos^2x -1)}{cos^2x}dx We know, cos 2x = 2cos2 x - 1
= ∫ 1/cos2x dx = ∫ sec2 x dx
= tan x + Constant
Question 6. Evaluate: ∫ elog√x /x dx
Solution:
∫ eloge √x /x dx = ∫√x/x dx
= ∫ x-1/2 dx = x-1/2 + 1/(-1/2 + 1) + Constant
= x1/2 /(1/2) + C
= 2√x + C