Unit 3: Fundamentals of integral calculus - Subjective Questions

An indefinite integral represents the family of all antiderivatives of a function. If , then

where is an arbitrary constant.

• The function is called the integrand.
• The symbol indicates that integration is performed with respect to .
• The constant is required because the derivative of every constant is zero.
• Therefore, functions such as and have the same derivative.

For example,

because for every constant .

The principal general rules are:

• Linearity: .
• Constant multiple rule: .
• Power rule: , provided .

Applying these rules,

The power rule states that

For , the denominator becomes , so the expression is undefined. Therefore, this case must be treated separately.

Since

we obtain

The absolute value is necessary because is defined on both intervals and , while the integrand is defined for every .

Use the standard formulas

and

Therefore,

Verification:

The derivative equals the original integrand, so the result is correct.

Integration by substitution reverses the chain rule of differentiation. If an integral contains a composite function together with the derivative , set

Then

The general procedure is:

• Select a suitable inner expression and set it equal to .
• Compute .
• Rewrite the complete integral in terms of and .
• Integrate with respect to .
• Substitute the original expression back for .

The method is particularly useful when the integrand contains a function and its derivative, possibly differing only by a nonzero constant factor.

Choose the expression inside the power as the substitution:

Differentiating,

Thus,

Using the power rule,

Substituting back gives

Differentiating the answer produces , confirming the result.

Let

Then

$$du=2x\

Indefinite integral: If , then is called an antiderivative of , and

where is an arbitrary constant.

Important rules:

• Linearity rule:
  
  where and are constants.
• Power rule: For ,
  
• For , the logarithmic rule is used:
  

The constant is necessary because the derivative of every constant is zero.

Integrate each term separately using the linearity rule:

Using standard formulas:

Therefore,

An indefinite integral represents a family of antiderivatives rather than a single function. Since the derivative of a constant is zero, functions differing only by a constant have the same derivative.

For example,

and

Thus, all functions of the form are antiderivatives of . Hence,

The constant accounts for every possible antiderivative and is called the constant of integration.

Integration by substitution is based on reversing the chain rule of differentiation.

Suppose an integral has the form

Choose

The integral becomes

which may be easier to evaluate. After integration, substitute back into the answer.

Substitution is useful when:

• The integrand contains a composite function and its derivative.
• A complicated algebraic or trigonometric expression can be replaced by one variable.
• Radicals such as occur.
• The integral contains forms such as .

For definite integrals, the limits must also be changed from -limits to the corresponding -limits.

Let

Then

Therefore,

Since ,

Substituting gives

Let

Thus,

Change the limits:

• When , .
• When , .

Therefore,

Hence,

Since ,

Use the substitution

Then

Also,

Therefore,

Since ,

Thus,

The product rule of differentiation is

Multiplying by gives

Integrating both sides,

Since , we obtain

Rearranging,

This is the integration-by-parts formula. It is especially useful for products involving algebraic, logarithmic, inverse trigonometric, exponential, or trigonometric functions.

Use

Choose

Then

Therefore,

Thus,

Factoring ,

Write the integral as

Choose

Then

Using integration by parts,

Therefore,

Hence,

Let

Then

Using integration by parts,

For the remaining integral, choose

so that

Thus,

Substituting this result,

Therefore,

Integration by substitution:

• It reverses the chain rule.
• It is suitable when the integrand contains a composite function together with its derivative.
• Its basic form is
  
• Example:
  

Integration by parts:

• It reverses the product rule.
• It is suitable mainly for products of functions.
• Its formula is
  
• Example:
  

Thus, substitution simplifies a composite function, whereas integration by parts transforms a product of functions.

Integration by partial fractions is used to integrate rational functions of the form

where and are polynomials and . If , polynomial division must be performed first.

Decomposition forms:

• Distinct linear factors:
  
• Repeated linear factor:
  
• Irreducible quadratic factor:
  
• If the irreducible quadratic is repeated, a linear numerator is included for every power of that factor.

After determining the unknown constants, each simpler fraction is integrated separately.