The integral I = ∫ (e(e^ sin x + sin x))cos xdx simplifies to (where, c is the constant of integration)

Correct answer: D. e^ e^ sin x + c The given integral is I = ∫ e^ esin x e^ sin x cos xdx Let e^ e^ sin x = t ⇒ (e^ esin x e^ sin x cos x)dx = dt ∴ I = ∫ 1dt = t + c ⇒ I = e^ e^ sin x + c

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Indefinite Integration

NTA Abhyas 2020 · Mathematics · Indefinite Integration

The integral I = ∫ (e(e^ sin x + sin x))cos xdx simplifies to...

The integral

I = ∫ \left(e\left(e^{sin x} + sin x\right)\right)cos xdx

simplifies to (where,

c

is the constant of integration)

Options

A $e^{sin x} + c$
B $e^{sin x + cos x} + c$
C $e^{e^{cos x}} + c$
D $e^{e^{sin x}} + c$

Correct answer

D. $e^{e^{sin x}} + c$

Step-by-step solution

The given integral is

I = ∫ e^{esin x}e^{sin x}cos xdx

Let

e^{e^{sin x}} = t

⇒ \left(e^{esin x}e^{sin x}cos x\right)dx = dt

∴ I = ∫ 1dt = t + c

⇒ I = e^{e^{sin x}} + c

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