Integral Calculus: Key Formulas and Techniques Explained

Study Notes

Integral Calculus: Exploring Key Formulas and Properties

Integral calculus is an essential branch of mathematics concerned with finding areas under curves or probabilities of events over time. As we delve into this subject matter, four important techniques and concepts emerge: integration by parts, substitution, definite integrals' properties, and integrating common trigonometric, exponential, and logarithmic functions.

Integration By Parts Formula

The integration by parts technique provides us with an antiderivative expression when two differentiable functions (u) and (v') are multiplied together. Its formula is given as:

[ \int udv = uv - \int vdu ]

While finding an appropriate pairing of (u) and (v) may seem challenging at first glance, it helps reduce complex integrals into simpler ones. For instance, consider evaluating (\int xe^x dx.) Here, let (u=x), so (du =dx,) and (v'=e^x), which implies (v=\int e^xdx=e^x). Applying the integration by parts formula yields:

[ \int xe^xdx = x(e^x)-\int e^xdx = xe^x-\int d(e^x)=xe^x-e^x+C]

Integration By Substitution Formula

Related to derivative rules like chain rule, the integration by substitution method lets you transform a given integral into another one involving a new variable (t). Essentially, you substitute (u) for a function of (x): (u = g(x)), where (du = g'(x)dx) and (dt = f'(x)dx). Then, rewrite the original integral using the new variables ((t)) and find its antiderivative. Once again, take the example (\int x\sin(x^2 + 1) dx). To apply the substitution method, choose (u = x^2 + 1 => du = 2x dx). Let (t = u => dt = 2x dx). Now, replace (u) and (dt) in the original integral:

[ \frac{1}{2} \int \sin t , dt = -\cos t + C = -\cos(x^2 + 1) + C]

Definite Integral Properties

Definite integrals involve computing specific intervals rather than arbitrary limits. Some fundamental properties include linearity, monotonicity, summability, and mean value theorem. These properties help solve various real-world problems such as determining the area between graphs and average values.

Trigonometric Function Integrals

Performing integrations of trigonometric functions relies heavily upon their periodic nature. Commonly encountered forms are:

[ \int sinx, dx = -cosx + C ] [ \int cosx, dx = sinx + C]

Furthermore, the following identities can simplify certain expressions:

[ \int sin^nxdx = \frac{-sin^{n-1}x \cdot cosx+(n-1)\int sin^{n-2} x \cdot cos^2 xdx } { n}] [ \int cos^nx, dx = \frac{cos^{n-1}x \cdot sinx-(n-1)\int cos^{n-2} x \cdot sin^2 xdx }{n }]

Exponential and Logarithmic Functions Integrals

For functions containing both exponentials and logs, follow these standard methods:

[ \int e^{ax}, dx = \frac{e^{ax}}{a}+C ] [ \int ln(x), dx = xln|x| - x + C ]

With practices and understanding of these key ideas, mastering integral calculus becomes an achievable goal!

Description

Explore the essential concepts in integral calculus including integration by parts, substitution, definite integrals' properties, and integrating trigonometric, exponential, and logarithmic functions. Mastering these key techniques is crucial for solving complex problems involving areas under curves and probabilities of events over time.