Apply the formula ∫eˣ(f(x) + f'(x))dx = eˣf(x) + C for instant solution.

Evaluate ∫eˣ(sin x + cos x) dx — Special eˣ Integral

Question

Evaluate:

\int e^x(\sin x + \cos x)\, dx

This appeared in JEE Main 2024 Shift 2 and is a classic one-step problem — if you know the formula.

Solution — Step by Step

The integrand is $e^x(\sin x + \cos x)$ . We need to check whether this fits the form $e^x[f(x) + f'(x)]$ .

Ask yourself: is $\cos x$ the derivative of $\sin x$ ? Yes. So let $f(x) = \sin x$ , which gives $f'(x) = \cos x$ .

The formula states:

\int e^x[f(x) + f'(x)]\, dx = e^x f(x) + C

This is a direct result — no substitution, no integration by parts needed.

Since $f(x) = \sin x$ :

\int e^x(\sin x + \cos x)\, dx = e^x \sin x + C

Final Answer: $e^x \sin x + C$

Why This Works

The formula $\int e^x[f(x) + f'(x)]\, dx = e^x f(x) + C$ comes from reversing the product rule. When we differentiate $e^x f(x)$ , we get $e^x f'(x) + e^x f(x) = e^x[f(x) + f'(x)]$ . So integration just undoes that.

This is why the formula requires exactly $f(x)$ and its derivative $f'(x)$ paired together. The $e^x$ acts as a multiplier throughout — it never changes form under differentiation, which is what makes this trick possible.

In board and JEE exams, this formula covers a whole family of questions. Any time you spot $e^x$ multiplied by two functions where one is the derivative of the other, you can write the answer in one step.

Alternative Method

We can verify using integration by parts twice — but honestly, this is the hard way. Use it only to build intuition or verify.

Let $I = \int e^x \sin x\, dx + \int e^x \cos x\, dx$ .

Integrate $\int e^x \cos x\, dx$ by parts:

\int e^x \cos x\, dx = e^x \cos x + \int e^x \sin x\, dx

Substituting back:

I = \int e^x \sin x\, dx + e^x \cos x + \int e^x \sin x\, dx

Now integrate $\int e^x \sin x\, dx$ by parts:

\int e^x \sin x\, dx = e^x \sin x - \int e^x \cos x\, dx

Substitute this in and the $\int e^x \cos x\, dx$ terms cancel cleanly, leaving $I = e^x \sin x + C$ .

In exams, never use this two-step parts approach unless the question explicitly asks you to verify. The formula method is 10 seconds; the parts method is 3 minutes with a real risk of sign errors.

Common Mistake

Students often write $\int e^x(\sin x + \cos x)\, dx = e^x(\sin x + \cos x) + C$ — thinking the whole bracket multiplies $e^x$ in the answer. This confuses the integrand pattern with the result pattern. The formula gives $e^x f(x)$ , not $e^x[f(x) + f'(x)]$ . Only $f(x) = \sin x$ appears in the answer, not $\cos x$ .

A quick self-check: differentiate your answer $e^x \sin x + C$ . You should get back $e^x \sin x + e^x \cos x = e^x(\sin x + \cos x)$ . If it matches the original integrand, you’re correct.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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