__Introduction__

__Introduction__

Antiderivatives, also known as indefinite integrals, are the inverse operation of derivatives. In other words, an antiderivative is the original function from which a derivative was derived.

For example, the derivative of f(x) = x² is f'(x) = 2x. Therefore, the antiderivative of 2x is f(x) = x². Antiderivatives can be used to find the area under a curve, and can also be used to solve differential equations.

### Integration Using Basic Integrals

In this specific exercise we are only going to solve some of the antiderivatives problems using basic integrals rules. “Remember to use the formulas provided below when solving the problems in this exercise. With these formulas, you’ll be able to solve all the questions quickly and accurately!”

### Exercise – 18.1

__Introduction__

__Introduction__

Antiderivatives, also known as indefinite integrals, are the inverse operation of derivatives. In other words, an antiderivative is the original function from which a derivative was derived.

For example, the derivative of f(x) = x² is f'(x) = 2x. Therefore, the antiderivative of 2x is f(x) = x². Antiderivatives can be used to find the area under a curve, and can also be used to solve differential equations.

### Integration by Substitution Method

In the preceding section we have seen that integrals can be integrated easily when they are in standard forms. A given integral may not always be in a standard form. So to reduce it to a standard form we have to change the variable into a new one by a suitable substitution. To make a proper choice one must have a lot of practice and skill. At the time of substitution, we express the given function x in term of the new variable y (say) and replace dx by (dy)/(dx)×dx.

### Exercise – 18.2

__Introduction__

__Introduction__

Antiderivatives, also known as indefinite integrals, are the inverse operation of derivatives. In other words, an antiderivative is the original function from which a derivative was derived.

For example, the derivative of f(x) = x² is f'(x) = 2x. Therefore, the antiderivative of 2x is f(x) = x². Antiderivatives can be used to find the area under a curve, and can also be used to solve differential equations.

### Integration Using Trigonometric Substitution

Here we shall consider integrals which involve a² – x², a² + x² or x² – a². It is quite obvious that the substitution x = asinθ turns a² – x² into a²cos²θ the substitution x = atanθ turns a² + x² into asec²θ and the substitution x = asecθ turns x² – a² into a²tan²θ. These substitutions will make the resulting function easily integrable. This can be better understood with some illustrations.

### Exercise – 18.3

__Introduction__

__Introduction__

### Integration by Parts

If a given function to be integrated is in the product form and it cannot be integrated either by reducing the integrand into the standard form or by substitution, we use the following rule known as the integration by parts.

The integral of the product of two functions = First function Integral of second

-Integral of (Derivative of first x Integral of second)

This is the formula of the integration of the product of two functions and is known as the “Integration by parts”. The successfulness of the use of the above formula depends upon the proper choice of the first function. The first function must be chosen such that its derivative reduces to a simple form and second function should be easily integrable.

### Exercise – 18.4

__Introduction__

__Introduction__

### Definite Integrals

Definite integral is a mathematical term used to denote a specific type of antiderivative. It is a tool used to calculate the area bounded by a curve, the volume of a solid of revolution, the length of a curve, and other such related quantities. It is a definite limit of a sum, known as a Riemann sum and denoted by an integral sign. It is defined as the limit of the sum of the products of the function values and the lengths of the subintervals of a partition of a given interval.

### Exercise – 18.5

__Introduction__

__Introduction__

### Definite Integral as an Area under the given curve

In The definite integral of a function f(x) is the area under the curve of f(x) between two points a and b. This area can be expressed as an integral, which is a calculus concept that allows us to calculate the area under a curve. It is calculated by taking the integral of the function between the two points of interest.

In the case of a definite integral, the area is bounded by a specific function and two points, typically a and b. The definite integral is then expressed as an expression of the form:

\int_a^b f(x) \, dx

This expression can be evaluated to calculate the area under the curve of f(x) between the two points a and b.