Integral Calculus Answers

Integrate with respect to x: ∫ e^5x sin⁡ 3x dx

a. 3e^5x/34(1/3 sin⁡ 3x-cos⁡ 3x)+c

b. 3e^5x/24(5/3 sin 3x-cos⁡ 3x)+c

c. 3e^5x/34(5/3 sin⁡ 3x-cos⁡ 3x)+c - yes* (please let me know if this is correct)

d. 3e^5x/34(1/3 sin⁡ 3x-cos⁡ 2x)+c

Integrate with respect to x: ∫ x^2 e^3x dx

a. e^3x/3(x^2-2x/3)+c

b. e^3x/3 (x^2-2x/3+2/9)+c - yes* (let me know if this is correct)

c. e^3x/3 (x-2x/3+2/9)+c

d. e^3x/3 (x^2-2x/3+2/7)+c

Integrate with respect to x: ∫ dx / (3 sin⁡x + 4 cos⁡x) dx

a. 13 ln⁡((1+(2 tan⁡x )/2)/(4-(2 tan⁡x)/2))+C

b. 15 ln⁡((1+(2 tan⁡x )/2)/(4-tan⁡x/2))+C

c. 1/4

d. 15 ln⁡((1+(2 tan⁡x )/2)/(4-(2 tan⁡x)/2))+C

Find the integral of x cos ax^2 dx with respect to x

a. cos^3x + c

b. sin^2x + c

c. sec^2x + 1

d. 1/2 a sin ax^2 + c

Find the area enclosed between the x-axis and the curve, if y = x^2 − 25 for (-5) is less than or equal to x less than or equal to 5.

a. 50

b. (-500)

c. 500

d. 25

Find the area enclosed between the x-axis and the curve, if y =x3 and the straight line y = x.

a. 4

b. 3x

c. 1/4

d. 12

Find the area enclosed between the x-axis and the curve, if y = x3 + 2x^2 + x + 1 between x = -1 and x = 2

a. 28

b. 14 1/4

c. 52

d. 2/5

Evaluate 3(x^2) -4x + 2 dx from first principles (upper bound: 4, lower bound: 2 )

Integrate with respect to x: ∫ x^2/((x-2)(x^2+1)) dx

4/5 ln⁡(x-2)+2/5 tan^(-1) x+c

4/5 ln(x-2)+1/5 ln⁡(x^2+1)+2/5 tan^(-1) x+c

4/5 ln(x-2)+1/10 ln⁡(x^2+1)+2/5 tan^(-1) x+c

4/5 ln(x-3)+1/10 ln⁡(x^2+1)+2/5 tan^(-1) x+c

Determine l = ∫ 2 cos⁡ x dx, given that l=7, when x=

cosx + sinx + c

2 sin x + 5

sin x + c

cosec x + c