Methods of Integration

The video tutorial covers various methods of integration, starting with basic techniques and moving to more advanced methods. It begins with an introduction to integration and its limitations, followed by a detailed explanation of integration by substitution, including examples involving trigonometric functions. The tutorial then generalizes the substitution method and solves several integration problems using this technique. Finally, it explores advanced integration techniques and alternative methods, providing a comprehensive understanding of integration strategies.

Which of the following is NOT a method of integration discussed in the introduction?

• Integration by differentiation

• Integration using partial fractions

• Integration by inspection

• Integration by substitution

What is the integral of tan x dx using substitution?

• log |sec x| + C

• log |cos x| + C

• log |sin x| + C

• log |tan x| + C

In the substitution method, what is the new variable t when integrating cot x dx?

• tan x

• cos x

• sec x

• sin x

What is the result of integrating sec x dx using substitution?

• log |sec x + tan x| + C

• log |cos x - sin x| + C

• log |sec x - tan x| + C

• log |cos x + sin x| + C

What is the integral of sine mx dx after substitution?

• m sin mx + C

• 1/m sin mx + C

• 1/m cos mx + C

• m cos mx + C

Which substitution is used for integrating 2x cos(x^2 - 1) dx?

• x^2 - 1 = t

• 2x = t

• cos(x^2 - 1) = t

• sin(x^2 - 1) = t

What is the integral of sec^2(t) * tan^4(t) dt after substitution?

• 5/2 sec^5(root x) + C

• 2/5 tan^5(root x) + C

• 5/2 tan^5(root x) + C

• 2/5 sec^5(root x) + C

What substitution is used for integrating sine of tan inverse of x?

• cos x = t

• tan x = t

• sin x = t

• tan inverse of x = t

What is the integral of sine^3(x) * cos^2(x) dx using substitution?

• 1/5 cos^3(x) - 1/3 cos^5(x) + C

• -1/3 cos^3(x) + 1/5 cos^5(x) + C

• 1/3 cos^3(x) - 1/5 cos^5(x) + C

• -1/5 cos^3(x) + 1/3 cos^5(x) + C

What is the integral of cos^2(x) dx using trigonometric identities?

• x/2 + 1/4 sin(2x) + C

• x/2 - 1/4 sin(2x) + C

• x + 1/2 sin(2x) + C

• x - 1/2 sin(2x) + C