Integration by parts — evaluate ∫x·eˣ dx step by step

Question

Evaluate $\displaystyle\int x \cdot e^x \, dx$ using integration by parts.

(NCERT Class 12, Chapter 7 — Integrals)

Solution — Step by Step

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

We need to choose which part of the integrand is $u$ and which is $dv$ .

The ILATE rule gives priority for choosing $u$ : Inverse trig > Logarithmic > Algebraic > Trigonometric > Exponential.

Here, $x$ is algebraic (A) and $e^x$ is exponential (E). Since A comes before E in ILATE:

u = x \implies du = dx

dv = e^x \, dx \implies v = e^x

\int x \cdot e^x \, dx = x \cdot e^x - \int e^x \, dx

= x \cdot e^x - e^x + C

\boxed{\int x \cdot e^x \, dx = e^x(x - 1) + C}

Why This Works

Integration by parts transfers the derivative from one function to another. By choosing $u = x$ , we differentiate it to get $du = dx$ (simpler). Meanwhile, $e^x$ integrates to itself, so nothing gets more complex. The net effect: the power of $x$ drops from 1 to 0, and the remaining integral $\int e^x \, dx$ is trivial.

If we had chosen $u = e^x$ and $dv = x \, dx$ , we’d get $v = x^2/2$ , making the new integral $\int \frac{x^2}{2} e^x \, dx$ — harder than what we started with. The ILATE rule prevents this bad choice.

Alternative Method — Using the formula for ∫eˣ·f(x)dx

There’s a direct result: $\int e^x[f(x) + f'(x)] \, dx = e^x \cdot f(x) + C$ .

We can write $xe^x = e^x[(x - 1) + 1] = e^x(x-1) + e^x$ .

Notice that if $f(x) = x - 1$ , then $f'(x) = 1$ , and $f(x) + f'(x) = x$ .

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

For JEE, the formula $\int e^x[f(x) + f'(x)] \, dx = e^x f(x) + C$ is extremely powerful. It bypasses integration by parts entirely. Whenever you see $e^x$ multiplied by a sum of a function and its derivative, apply this directly. This saves significant time in competitive exams.

Common Mistake

Students sometimes apply ILATE mechanically and choose $u = e^x$ (thinking “E comes last, so it should be $dv$ ”). ILATE tells you which function to pick as $u$ (the one that appears earlier in the list), not $dv$ . Algebraic ( $x$ ) comes before Exponential ( $e^x$ ) in ILATE, so $u = x$ . Getting this backwards leads to a more complicated integral instead of a simpler one.

Question

Solution — Step by Step

Why This Works

Alternative Method — Using the formula for ∫eˣ·f(x)dx

Common Mistake

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