Applications of integrals #10 - Questions and Answers

Question 1 Solve the integral. ∫ 3sec(2x - 1)tan(2x - 1)dx Answer Step 1 Given integral is ∫ 3sec(2x - 1)tan(2x - 1)dx We will use the following substitution to evaluate the integral. 2x-1=t 2dx=dt dx = 1/2 dt Step 2 Therefore, we have ∫ 3sec(2x - 1)tan(2x - 1)dx = ∫ 3/2 sec t tan tdt We know that integral of secxtanxdx is secx+c Therefore, ∫ 3sec(2x - 1)tan(2x - 1)dx = 3/2 ∫ sec t tan tdt We know that integral of secxtanxdx is secx+c Therefore, ∫ 3sec(2x - 1)tan(2x - 1)dx = 3/2 sect + c ∫ 3sec(2x - 1)tan(2x - 1)dx = 3/2 sec(2x-1) + c Step 3 Ans: ∫ 3sec(2x - 1)tan(2x - 1)dx = 3/2 sec(2x-1) + c Question 2 Evaluate the integral: ∫ secx(sec x + tanx)dx Answer 1 Step 1 To find the below integral. ∫ secx(sec x + tanx)dx Step 2 Using u substitution method u = sec x + tanx ⇒ du = (sec²x + tan x sec x)dx ⇒ du = secx(secx + tanx)dx ⇒ du = u sec xdx ⇒ secxdx = du/u ∫ secx(sec x + tanx)dx = ∫ u * du/u ∫ secx(sec x + tanx)dx = ∫ du ∫ secx(sec x + tanx)dx = u + C ∫ secx(sec x + tanx)dx = sec x + tanx + C Answer 2 Step 1: Expand. ∫ sec²x + secxtanxdx Step 2: Use Sum Rule: ∫ f(x) + g(x)dx = ∫ f(x)dx + ∫ g(x)dx. ∫ sec²xdx + ∫ secxtanxdx Step 3: The derivative of tan x is sec²x. tan x + ∫ secxtanxdx Step 4: The derivative of sec x is secxtan x. tan x + sec x Step 5: Add constant. tan x + sec x + C Question 3 Evaluate the indefinite integral. ∫ (cot x)ln(sin x)dx Answer Step 1 We must assess the integral ∫ (cotx) ln(sin x) dx ...(1) let cotx = z log (sinx) = dz/dx Step 2 substituting it in the equation (1) we get ∫ z dz integrating it we get z²/2 adding z's value, we obtain cot²x/2 therefore the answer of ∫ (cotx) ln (sin x) dx is cot²x/2 Answer 3 Step 1: Remove parentheses. ∫ cotx ln(sin x) dx Step 2: Use Integration by Parts on ∫ cotx ln(sin x)dx Let u = ln (sin x), dv = cot x dx, du = cos x/sin x dx = cotx dx, v = ∫ cotx dx = ln(sin x) Step 3: Substitute the above into uv - ∫ vdu. = ln(sin x) ln (sin x) - ∫ ln (sin x) cot x dx Step 4: Use Integration by Substitution on ∫ ln(sin x) cot x dx Let u = ln(sin x), du = cotx dx ∫ ln u du Step 6: Use Integration by Substitution. Let w = ln u, dw = 1/u du Step 7: Using w and dw above, rewrite ∫ ln u du. ∫ w dw Step 8: Use Power Rule: ∫ xⁿ dx = x^(n+1)/(n+1) + C. = w²/2 Step 9: Substitute w = ln u back into the original integral. = (ln u)²/2 Step 10: Substitute u = sin x back into the original integral. = (ln (sin x))²/2 Step 11: Rewrite the integral with the completed substitution. = (ln(sin x))²/2 Step 12: Add constant. = (ln(sin x))²/2 + C Question 4 Evaluate the definite integral using integration by parts. ∫(from 0 to 2) z(z - 2)⁴dz Answer 1 Step 1 We need to Evaluate the definite integral using integration by parts of the following integral. ∫(from 0 to 2) z(z - 2)⁴dz Note: Integration by parts: ∫(from a to b) u dv/dz dz = [uv](from a to b) - ∫(from a to b) v du/dz dz Step 2 So, For the given integral, ∫(from 0 to 2) z(z - 2)⁴dz = [z(z-2)⁵/5](from 0 to 2) - ∫(from 0 to 2) (z-2)⁵/5 * d/dz (z) dz = [z(z-2)⁵/5](from 0 to 2) - ∫(from 0 to 2) (z-2)⁵/5 dz = [z(z-2)⁵/5](from 0 to 2) - [(z-2)⁶/(56)](from 0 to 2) = (0) - 0 - ((0-(-2)⁶/(56)) = -64/30 = -32/15 Answer: In the exact form: ∫(from 0 to 2) z(z - 2)⁴dz = [z(z-2)⁵/5](from 0 to 2) - ∫(from 0 to 2) ((z-2)⁵/5 * d/dz (z)) dz = -32/15 And the value of the integral is ∫(from 0 to 2) z(z - 2)⁴dz = -32/15 Answer 2 Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then: ∫(from a to b) f(x)dx = F(x) = F(b) - F(a) Step 2: In this case, f(z) = z(x - 2)⁴. Find its integral. = (z(z-2)⁵/5) - ∫ (z-2)⁵/5 dz = z(z-2)⁵/5 - (z-2)⁶/30 Step 3: Since F(z) = F(b) - F(a), expand the above into F(2)-F(0): = ((2(2-2)⁵/5) - ((2-2)⁶/30)) - ((0(0-2)⁵/5) - ((0-2)⁶/30)) Step 4: Simplify. = 32/15 Question 5 Evaluate the integral. ∫ 7x ln x dx Answer 1 Step 1 Consider the integral ∫ 7x ln x dx Step 2 Integration by parts ∫ uv' dx = u ∫ v dx - ∫ u'(∫ v dx) dx ∫ 7x ln x dx Here u = ln x v'= 7x Now ∫ 7x ln x dx = ln x ∫ 7x dx - ∫ (ln x)' (∫ 7x dx)dx = ln x [7x²/2] - ∫ (1/x)(7x²/2) dx = 7x²ln x / 2 - 7/2 ∫ x dx = 7x²ln x/ 2 - 7/2 (x²/2) + C = 7x² ln x / 2 - 7x²/4 = x²/2(7ln x - 7/2) Answer: ∫ 7x ln x dx = 7x² ln x / 2 - 7x²/4 + C Answer 2 Step 1: Use Constant Factor Rule: ∫cf(x)dx = c ∫f(x)dx. ∫ 7x ln xdx = 7 ∫x ln x dx Step 2: Use Integration by Parts on ∫ x ln x dx. Let u = ln x, dv = x dx, du = 1/x dx, v = x²/2 Step 3: Substitute the above into uv - ∫ vdu. = 7(x²ln x/2 - ∫ x²/2 * 1/x dx) Step 4: Use Constant Factor Rule: ∫cf(x)dx = c ∫f(x)dx. = 7(x²ln x/2 - 1/2 ∫ x dx) Step 5: Use Power Rule: ∫ xⁿ dx = x^(n+1)/(n+1) + C. = 7(x²ln x/2 - 1/2 x²/2) Step 6: Add constant. = 7(x²ln x/2 - x²/4) + C