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Chapter 7- Integration

● Indefinite and definite integrals, including the area under a graph● Integrating standard functions● Integrating by inspection, by substitution, and by parts● Convergence of definite integrals● Multiple integrals● Numerical methods for integration● Taylor series and Maclaurin series● Simple ordinary differential equations.

INTEGRALS

Integration and differentiation are connected in that they are the “reverse” of each other.

Integrals with limits are called definite integrals.

INTEGRALS

INTEGRATION OF STANDARD FUNCTIONS

INTEGRATION OF STANDARD FUNCTIONS

INTEGRATION OF STANDARD FUNCTIONS

INTEGRATION OF STANDARD FUNCTIONS

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

Integration by inspection

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

Integration using partial fractions

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

Integration by substitution (change of variable)

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

Integration by parts

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

Differentiating an integral (Leibniz’s formula)

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

INTEGRATION BY INSPECTION, SUBSTITUTION & PARTS

CONVERGENCE

CONVERGENCE

CONVERGENCE

CONVERGENCE

DOUBLE INTEGRALS

DOUBLE INTEGRALS

DOUBLE INTEGRALS

DOUBLE INTEGRALS

DOUBLE INTEGRALS

DOUBLE INTEGRALS

Swapping the order of integration

DOUBLE INTEGRALS

DOUBLE INTEGRALS

NUMERICAL METHODS FOR INTEGRATION

The trapezium rule

THE TRAPEZIUM RULE

THE TRAPEZIUM RULE

TAYLOR & MACLAURIN SERIES

Taylor series

TAYLOR AND MACLAURIN SERIES

TAYLOR AND MACLAURIN SERIES

TAYLOR AND MACLAURIN SERIES

TAYLOR AND MACLAURIN SERIES

TAYLOR AND MACLAURIN SERIES

TAYLOR AND MACLAURIN SERIES

TAYLOR AND MACLAURIN SERIES

Maclaurin series

TAYLOR AND MACLAURIN SERIES

TAYLOR AND MACLAURIN SERIES

DIFFERENTIAL EQUATIONS

Solution by direct integration

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS

Solution by separation of variables

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS

Solution by integrating factor

DIFFERENTIAL EQUATIONS