Explain integration by parts with formula

Integration by parts is the reverse of the product rule for differentiation. It helps you integrate products of two functions. The formula is:

$$\int u\,dv = uv - \int v\,du$$

Here, $u$ is the part you differentiate, and $dv$ is the part you integrate. You must choose $u$ and $dv$ wisely. The rule to remember is ILATE (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential) – choose $u$ in that order of priority.

Steps: 1. Choose $u$ and $dv$ from your integral. 2. Differentiate $u$ to get $du$. 3. Integrate $dv$ to get $v$. 4. Plug into the formula and simplify.

Micro-example: Find $\int x\,e^x\,dx$. Let $u = x$ (Algebraic, higher priority than Exponential), so $du = dx$. Let $dv = e^x\,dx$, so $v = e^x$. Then:

$$\int x\,e^x\,dx = x\,e^x - \int e^x\,dx = x\,e^x - e^x + C = e^x(x - 1) + C$$

Key tips: If you get a simpler integral after applying, you chose correctly. If it gets harder, swap $u$ and $dv$. For definite integrals, apply limits to the final expression. Sometimes you need to use parts twice (e.g., $\int x^2\sin x\,dx$). Practice makes perfect – solve at least 10 problems to get the hang of it.

Remember: integration by parts is your best friend for products like $x\ln x$, $x\sin x$, $x\cos x$, and $\ln x$ (by taking $dv = dx$). Always add the constant $C$ for indefinite integrals. This technique is a must for board exams and JEE – so master it well.