Teach me integration- indefinite definite, integral formulas, substitution, trigonometric substitutions, integration by parts, standard ntegrals, use of partial frations etc for BCA

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Integration is one of the fundamental concepts in calculus. It serves two main purposes: finding antiderivatives and calculating areas under curves. There are two types of integrals. The indefinite integral represents a family of functions and includes a constant of integration. The definite integral gives us a specific numerical value representing the area under the curve between two limits. To master integration, you must memorize the standard formulas. The power rule states that the integral of x to the n equals x to the n plus one divided by n plus one, plus the constant of integration. For the natural logarithm, the integral of one over x equals natural log of absolute value x plus C. The exponential function e to the x integrates to itself. For trigonometric functions, sine integrates to negative cosine, cosine integrates to sine, and secant squared integrates to tangent. The substitution method is used when the integrand contains a function and its derivative. The key is to identify a suitable substitution u equals g of x, where g prime of x also appears in the integrand. In our example, we have 2x times x squared plus 1 to the third power. We let u equal x squared plus 1, so du equals 2x dx. This transforms our integral into u cubed du, which integrates to u to the fourth over 4. Substituting back gives us x squared plus 1 to the fourth power over 4 plus C. Integration by parts is used for products of functions. The formula is the integral of u dv equals uv minus the integral of v du. To choose u, use the LIATE rule: prioritize Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, then Exponential functions. In our example, x times e to the x, we choose u equals x because it's algebraic, and dv equals e to the x dx because it's exponential. This gives us du equals dx and v equals e to the x. Applying the formula yields x e to the x minus the integral of e to the x dx, which simplifies to x e to the x minus e to the x plus C. To master integration for your BCA studies, focus on these key areas. First, memorize the standard integration formulas for power functions, exponentials, and trigonometric functions. Second, use substitution when you see a function and its derivative appearing together in the integrand. Third, apply integration by parts for products of functions, using the LIATE rule to choose which part to differentiate. Fourth, practice trigonometric substitutions for expressions involving square roots. Finally, learn to decompose rational functions using partial fractions. With these techniques, you'll be well-equipped to handle most integration problems in your coursework.