If the integral Ι = ∫ e^ sin x (cos x ⋅ x^ 2 + 2x)dx =e^ f(x) g(x) + C (where, C is the constant of integration), then the number of solution(s) of f (x) = g (x) is/are

Correct answer: B. 2 As, Ι = ∫ e^ f(x) (f^ ′ (x)g(x) + g^ ′ (x))dx = e^ f(x) g^ (x) + C Thus, ∫ e^ sin x (cos x ⋅ x^ 2 + 2x)dx = e^ sin x ⋅ x^ 2 + C i.e. f(x) = sin x g(x) = x^ 2 Which intersect at ‘2’ points.

NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If the integral Ι = ∫ e^ sin x (cos x ⋅ x^ 2 + 2x)dx =e^ f(x) g(x)...

If the integral

Ι = ∫ e^{sin x}\left(cos x ⋅ x^{2} + 2x\right)dx =e^{f\left(x\right)}g\left(x\right) + C

(where,

C

is the constant of integration), then the number of solution(s)

of f \left(x\right) = g \left(x\right) is/are

If the integral Ι = ∫ e^ sin x (cos x ⋅ x^ 2 + 2x)dx =e^ f(x) g(x) + C (where, C is the constant of integration), then

Options

A $0$
B $2$
C $4$
D $6$

Correct answer

B. $2$

Step-by-step solution

As,

\text{Ι} = ∫ e^{f\left(x\right)} \left(f^{′} \left(x\right)g\left(x\right) + g^{′} \left(x\right)\right)dx = e^{f\left(x\right)} g^{\left(x\right)} + C

Thus,

∫ e^{sin x}\left(cos x ⋅ x^{2} + 2x\right)dx = e^{sin x} ⋅ x^{2} + C

i.e.

f\left(x\right) = sin x g\left(x\right) = x^{2}

Which intersect at

‘2’

points.

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