Integral Calculus - Problem on integral of derivative form

• In this lecture, we will discuss problems on integrating the derivative form.

Problem 1

Consider the function f(x) = x^3 + 3x^2 + 2x + 1. Find F(x), where F(x) is the antiderivative of f(x).

• Solution:

  • We can find the antiderivative of f(x) by using the power rule for integration:

    • ∫(x^n) dx = (1/(n+1)) * x^(n+1)

  • Applying the power rule, we get:

    • F(x) = (1/4) * x^4 + (1) * x^3 + (1/2) * x^2 + (1) * x + C

• Therefore, the antiderivative of f(x) is F(x) = (1/4) * x^4 + x^3 + (1/2) * x^2 + x + C.

Problem 2

Evaluate ∫(4x^3 + 2x^2 - 5x + 3) dx.

• Solution:

  • To find the integral of a polynomial function, we apply the power rule for integration to each term and sum the results.

  • Applying the power rule, we get:

    • ∫(4x^3) dx = (4/4) * x^4 = x^4

    • ∫(2x^2) dx = (2/3) * x^3

    • ∫(-5x) dx = (-5/2) * x^2

    • ∫(3) dx = 3x

  • Adding the results, we get:

    • ∫(4x^3 + 2x^2 - 5x + 3) dx = x^4 + (2/3) * x^3 - (5/2) * x^2 + 3x + C

• Therefore, the value of the integral is x^4 + (2/3) * x^3 - (5/2) * x^2 + 3x + C.

Problem 3

Find the value of ∫(2cos(2x)) dx.

• Solution:

  • To integrate a cosine function, we use the power rule for integration.

  • Applying the power rule, we get:

    • ∫(cos(x)) dx = sin(x)

  • However, in this case, we have ∫(2cos(2x)) dx, so we need to apply a substitution.

  • Let u = 2x. Then, du = 2 dx.

  • Substituting the values, we get:

    • ∫(cos(2x)) dx = (1/2) * ∫(cos(u)) du = (1/2) * sin(u) = (1/2) * sin(2x)

• Therefore, the value of the integral is (1/2) * sin(2x) + C.

Problem 4

Evaluate ∫(e^(3x)) dx.

• Solution:

  • To integrate the exponential function e^(kx), we apply the natural logarithm property of integrals.

  • Applying the property, we get:

    • ∫(e^(kx)) dx = (1/k) * e^(kx)

  • In this case, k = 3, so we have:

    • ∫(e^(3x)) dx = (1/3) * e^(3x)

• Therefore, the value of the integral is (1/3) * e^(3x) + C.

Problem 5

Find ∫(x * ln(x)) dx.

• Solution:

  • To integrate a product of functions, we use the integration by parts method.

  • The integration by parts formula is:

    • ∫(u * dv) = u * v - ∫(v * du)

  • In this case, let u = ln(x) and dv = x dx.

  • Differentiating u and integrating dv, we get:

    • du = (1/x) dx and v = (1/2) * x^2

  • Applying the integration by parts formula, we get:

    • ∫(x * ln(x)) dx = (1/2) * x^2 * ln(x) - ∫((1/2) * x^2 * (1/x)) dx

    • Simplifying, we have:

      • ∫(x * ln(x)) dx = (1/2) * x^2 * ln(x) - (1/2) * ∫(x) dx

    • Integrating the remaining term, we get:

      • ∫(x) dx = (1/2) * x^2

    • Substituting the values, we have:

      • ∫(x * ln(x)) dx = (1/2) * x^2 * ln(x) - (1/2) * (1/2) * x^2 + C

• Therefore, the value of the integral is (1/2) * x^2 * ln(x) - (1/4) * x^2 + C.

Problem 6

Evaluate ∫(cot^2(x)) dx.

• Solution:

  • To integrate the cotangent squared function, we use a trigonometric identity.

  • The trigonometric identity is:

    • cot^2(x) = csc^2(x) - 1

  • Applying the identity, we get:

    • ∫(cot^2(x)) dx = ∫(csc^2(x) - 1) dx

    • Breaking the integral into two parts, we have:

      • ∫(csc^2(x)) dx - ∫(1) dx

  • Integrating the first part, we get:

    • ∫(csc^2(x)) dx = -cot(x)

  • Integrating the second part, we get:

    • ∫(1) dx = x

  • Adding the results, we have:

    • ∫(cot^2(x)) dx = -cot(x) - x + C

• Therefore, the value of the integral is -cot(x) - x + C.

Problem 7

Find ∫(1/(x^2 + 9)) dx.

• Solution:

  • To integrate the rational function, we can use the substitution method.

  • Let u = x^2 + 9. Then, du = 2x dx.

  • Rearranging the equation, we get:

    • (1/2) * du = x dx

  • Substituting the values, we have:

    • ∫(1/(x^2 + 9)) dx = (1/2) * ∫(1/u) du

    • Simplifying, we get:

      • ∫(1/(x^2 + 9)) dx = (1/2) * ln|u| + C

      • Since u = x^2 + 9, we have:

        • ∫(1/(x^2 + 9)) dx = (1/2) * ln|x^2 + 9| + C

• Therefore, the value of the integral is (1/2) * ln|x^2 + 9| + C.

Problem 8

Evaluate ∫(sec^2(x) * tan(x)) dx.

• Solution:

  • To integrate the product of secant squared and tangent functions, we can use a substitution.

  • Let u = sec(x). Then, du = sec(x) * tan(x) dx.

  • Substituting the values, we have:

    • ∫(sec^2(x) * tan(x)) dx = ∫(du) = u = sec(x)

• Therefore, the value of the integral is sec(x) + C.

Problem 9

Find ∫(e^x - x^2 + 5x) dx.

• Solution:

  • To find the integral of a combination of functions, we apply the power rule for integration to each term and sum the results.

  • Applying the power rule, we get:

    • ∫(e^x) dx = e^x

    • ∫(x^2) dx = (1/3) * x^3

    • ∫(5x) dx = (5/2) * x^2

  • Adding the results, we get:

    • ∫(e^x - x^2 + 5x) dx = e^x - (1/3) * x^3 + (5/2) * x^2 + C

• Therefore, the value of the integral is e^x - (1/3) * x^3 + (5/2) * x^2 + C.

Problem 10

Evaluate ∫(x * e^(2x)) dx.

• Solution:

  • To integrate a product of functions involving the exponential function, we can use the integration by parts method.

  • The integration by parts formula is:

    • ∫(u * dv) = u * v - ∫(v * du)

  • In this case, let u = x and dv = e^(2x) dx.

  • Differentiating u and integrating dv, we get:

    • du = dx and v = (1/2) * e^(2x)

  • Applying the integration by parts formula, we get:

    • ∫(x * e^(2x)) dx = (1/2) * x * e^(2x) - ∫((1/2) * e^(2x)) dx

    • Simplifying, we have:

      • ∫(x * e^(2x)) dx = (1/2) * x * e^(2x) - (1/2) * ∫(e^(2x)) dx

    • Integrating the remaining term, we get:

      • ∫(e^(2x)) dx = (1/2) * e^(2x)

    • Substituting the values, we have:

      • ∫(x * e^(2x)) dx = (1/2) * x * e^(2x) - (1/2) * (1/2) * e^(2x) + C

• Therefore, the value of the integral is (1/2) * x * e^(2x) - (1/4) * e^(2x) + C.

• Problem: Find the antiderivative of f(x) = 6x^2 - 4x + 2.
• Solution:
  • Apply the power rule for integration to each term:
    • ∫(6x^2) dx = 2x^3
    • ∫(-4x) dx = -2x^2
    • ∫(2) dx = 2x
  • Summing the results, we get the antiderivative:
    • F(x) = 2x^3 - 2x^2 + 2x + C

• Problem: Evaluate ∫(sin^2(x) + cos^2(x)) dx.
• Solution:
  • Apply the Pythagorean identity sin^2(x) + cos^2(x) = 1:
    • ∫(1) dx = x
  • Therefore, ∫(sin^2(x) + cos^2(x)) dx = x + C

• Problem: Find the integral of f(x) = 1/x.
• Solution:
  • Apply the natural logarithm property of integrals:
    • ∫(1/x) dx = ln|x|
  • Therefore, the integral of f(x) is ln|x| + C

• Problem: Evaluate ∫(3x^2 + 2x - 4) dx.
• Solution:
  • Apply the power rule for integration to each term:
    • ∫(3x^2) dx = x^3
    • ∫(2x) dx = x^2
    • ∫(-4) dx = -4x
  • Summing the results, we get the integral value:
    • ∫(3x^2 + 2x - 4) dx = x^3 + x^2 - 4x + C

• Problem: Find the integral of e^(3x).
• Solution:
  • Apply the natural logarithm property of integrals:
    • ∫(e^(3x)) dx = (1/3) * e^(3x)
  • Therefore, the integral of e^(3x) is (1/3) * e^(3x) + C

• Problem: Evaluate ∫(2sec^2(x) + 2tan(x)) dx.
• Solution:
  • Apply the integral of sec^2(x):
    • ∫(2sec^2(x)) dx = 2tan(x)
  • Apply the integral of tan(x):
    • ∫(2tan(x)) dx = 2ln|sec(x)|
  • Therefore, the integral is 2tan(x) + 2ln|sec(x)| + C

• Problem: Find ∫(5sin(x) + 3cos(x)) dx.
• Solution:
  • Apply the integral of sin(x):
    • ∫(5sin(x)) dx = -5cos(x)
  • Apply the integral of cos(x):
    • ∫(3cos(x)) dx = 3sin(x)
  • Therefore, the integral is -5cos(x) + 3sin(x) + C

• Problem: Evaluate ∫(4x^2 - 5x + 1/x) dx.
• Solution:
  • Apply the power rule for integration to the polynomial terms:
    • ∫(4x^2) dx = (4/3) * x^3
    • ∫(-5x) dx = (-5/2) * x^2
  • Apply the natural logarithm property for the rational term:
    • ∫(1/x) dx = ln|x|
  • Summing the results, we get the integral value:
    • ∫(4x^2 - 5x + 1/x) dx = (4/3) * x^3 - (5/2) * x^2 + ln|x| + C

• Problem: Find the integral of f(x) = e^(-x).
• Solution:
  • Apply the natural logarithm property of integrals:
    • ∫(e^(-x)) dx = -e^(-x)
  • Therefore, the integral of f(x) is -e^(-x) + C

• Problem: Evaluate ∫(7sin(x) - 3cos(x)) dx.
• Solution:
  • Apply the integral of sin(x):
    • ∫(7sin(x)) dx = -7cos(x)
  • Apply the integral of cos(x):
    • ∫(-3cos(x)) dx = -3sin(x)
  • Therefore, the integral is -7cos(x) - 3sin(x) + C

Integral Calculus - Problem on integral of derivative form

Slide 21:

• Problem: Find the integral of f(x) = ln(x). Solution:
• Apply the power rule for integration with a natural logarithm:
  • ∫(ln(x)) dx = x ln(x) - ∫(x * (1/x)) dx = x ln(x) - ∫dx = x ln(x) - x + C

Slide 22:

• Problem: Evaluate ∫(2x + 5) dx. Solution:
• Apply the power rule for integration to each term:
  • ∫(2x) dx = x^2
  • ∫(5) dx = 5x
• Summing the results, we get:
  • ∫(2x + 5) dx = x^2 + 5x + C

Slide 23:

• Problem: Find the integral of f(x) = e^(2x). Solution:
• Apply the natural logarithm property of integrals:
  • ∫(e^(2x)) dx = (1/2) * e^(2x)

Slide 24:

• Problem: Evaluate ∫(3sin(x) + 4cos(x)) dx. Solution:
• Apply the integral of sin(x):
  • ∫(3sin(x)) dx = -3cos(x)
• Apply the integral of cos(x):
  • ∫(4cos(x)) dx = 4sin(x)
• Summing the results, we get:
  • ∫(3sin(x) + 4cos(x)) dx = -3cos(x) + 4sin(x) + C

Slide 25:

• Problem: Find the integral of f(x) = 1/(x^2 + 1). Solution:
• Apply the arctan property of integrals:
  • ∫(1/(x^2 + 1)) dx = arctan(x) + C

Slide 26:

• Problem: Evaluate ∫(2sec^2(x) - 3csc^2(x)) dx. Solution:
• Apply the integral of sec^2(x):
  • ∫(2sec^2(x)) dx = 2tan(x)
• Apply the integral of csc^2(x):
  • ∫(-3csc^2(x)) dx = -3cot(x)
• Summing the results, we get:
  • ∫(2sec^2(x) - 3csc^2(x)) dx = 2tan(x) - 3cot(x) + C

Slide 27:

• Problem: Find the integral of f(x) = x^2 * e^(3x). Solution:
• Apply