Title: Indefinite Integral - Examples of specific types of problems
- Introduction to indefinite integrals
- Importance of finding antiderivatives
- Overview of specific types of problems covered in this lecture
Type 1: Basic Power Rule
- Formula: ∫xn,dx=n+1xn+1+C
- Examples:
- ∫x3,dx=4x4+C
- ∫x−2,dx=−x1+C
Type 2: Exponential Functions
- Formula: ∫ex,dx=ex+C
- Examples:
- ∫e2x,dx=2e2x+C
- ∫e−3x,dx=−31e−3x+C
Type 3: Trigonometric Functions
- Formula: ∫sin(x),dx=−cos(x)+C
- Examples:
- ∫sin(2x),dx=−2cos(2x)+C
- ∫cos(3x),dx=3sin(3x)+C
Type 4: Logarithmic Functions
- Formula: ∫x1,dx=ln∣x∣+C
- Examples:
- ∫2x1,dx=ln∣2x∣+C
- ∫x+11,dx=ln∣x+1∣+C
Type 5: Rational Functions
- Formula: Integration by parts or partial fraction decomposition may be required
- Examples:
- ∫x2+11,dx (partial fraction decomposition needed)
- ∫xln(x),dx (integration by parts needed)
Type 6: Trigonometric Substitution
- Formula: Used for integrating expressions containing square roots and trigonometric functions
- Substitutions:
- x=sin(t)
- x=cos(t)
- x=tan(t)
- Examples:
- ∫1−x2dx
- ∫1+x2,dx
Type 7: Hyperbolic Functions
- Formula: ∫sinh(x),dx=cosh(x)+C
- Examples:
- ∫sinh(2x),dx=2cosh(2x)+C
- ∫cosh(3x),dx=3sinh(3x)+C
Type 8: Integration by Parts
- Formula: ∫u,dv=uv−∫v,du
- Examples:
- ∫xcos(x),dx
- ∫x2ex,dx
Type 9: Improper Integrals
- Formula: Evaluate limits as the interval approaches infinity or a point of discontinuity
- Examples:
- ∫0∞e−x,dx
- ∫−∞∞1+x21,dx
Type 10: Integration by Substitution
- Formula: ∫f(g(x))g’(x),dx=∫f(u),du, where u=g(x)
- Examples:
- ∫2xcos(x2),dx
- ∫ex,dx
Type 11: Integration of Composite Functions
- Formula: ∫f(g(x)),dx=F(g(x)),dx
- Examples:
- ∫1−sin(x),dx
- ∫1−x21,dx
Type 12: Integration of Rational Functions
- Formula: Use partial fraction decomposition to simplify the expression
- Examples:
- ∫x2−4x+42x+3,dx
- ∫x−1x3−1,dx
Type 13: Integration of Trigonometric Products
- Formula: Use product-to-sum identities or trigonometric identities to simplify the expression
- Examples:
- ∫sin2(x)cos3(x),dx
- ∫tan(x)sec2(x),dx
Type 14: Integration of Exponential Functions
- Formula: Apply appropriate substitution or integration by parts
- Examples:
- ∫excos(x),dx
- ∫e2xsin(3x),dx
Type 15: Integration of Logarithmic Functions
- Formula: Integration by parts or substitution may be required
- Examples:
- ∫xln(x),dx
- ∫sin−1(x),dx
Type 16: Integration of Hyperbolic Functions
- Formula: Use appropriate hyperbolic identities or integration techniques
- Examples:
- ∫sinh3(x),dx
- ∫cosh(2x)sinh(2x),dx
Type 17: Integration of Trigonometric Powers
- Formula: Use various trigonometric identities and manipulation techniques
- Examples:
- ∫sin3(x)cos2(x),dx
- ∫tan2(x)sec3(x),dx
Type 18: Inverse Trigonometric Functions
- Formula: Use appropriate substitution or trigonometric identities
- Examples:
- ∫1−x21,dx
- ∫1+x21,dx
Type 19: Integration of Powers of Trigonometric Functions
- Formula: Simplify using power reduction formulas or appropriate manipulation
- Examples:
- ∫sin5(x),dx
- ∫cos4(x),dx
Type 20: Integration of Inverse Trigonometric Functions
- Formula: Use appropriate substitution or trigonometric identities
- Examples:
- ∫1−x21,dx
- ∫1+x21,dx
Type 21: Integration of Powers of Trigonometric Functions
- Formula: Simplify using power reduction formulas or appropriate manipulation
- Examples:
- ∫sin5(x),dx
- ∫cos4(x),dx
Type 22: Integration of Rational Expressions with Trigonometric Functions
- Formula: Use trigonometric identities or appropriate substitution
- Examples:
- ∫cos4(x)sin2(x),dx
- ∫tan2(x)sec(x),dx
Type 23: Integration involving Absolute Values
- Formula: Split into cases and solve each separately
- Examples:
- ∫∣x−2∣,dx
- ∫∣x2−4∣,dx
Type 24: Integration of Hyperbolic Trigonometric Functions
- Formula: Use corresponding hyperbolic identities or appropriate techniques
- Examples:
- ∫sinh2(x),dx
- ∫cosh(2x)sinh(3x),dx
Type 25: Integration of Logarithmic-exponential Functions
- Formula: Apply integration by parts or substitution
- Examples:
- ∫xln(ex),dx
- ∫exln(x),dx
Type 26: Integration of Trigonometric Functions raised to a Power
- Formula: Use reduction formula or appropriate manipulation techniques
- Examples:
- ∫sin3(x)cos2(x),dx
- ∫tan2(x)sec3(x),dx
Type 27: Integration of Hyperbolic Functions raised to a Power
- Formula: Use reduction formula or appropriate manipulation techniques
- Examples:
- ∫sinh3(x)cosh2(x),dx
- $\int \tanh^2(x) \sech^3(x) , dx$
Type 28: Integration of Arc Length
- Formula: Apply integration techniques to calculate arc length
- Examples:
- ∫1+(cos(x))2,dx
- ∫1+(sinh(x))2,dx
Type 29: Integration of Logarithmic Trigonometric Functions
- Formula: Use substitution or appropriate manipulation techniques
- Examples:
- ∫ln(sin(x)),dx
- ∫ln(cos(x)),dx
Title: Indefinite Integral - Examples of specific types of problems Introduction to indefinite integrals Importance of finding antiderivatives Overview of specific types of problems covered in this lecture