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NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If I _ n = ∫ x^ n e^ 6x dx , then the expression 6 I _ 10 + 10 I _...

If

{I}_{n} = ∫ x^{n}e^{6x}dx

, then the expression

6{I}_{10} + 10{I}_{9}

simplifies to (where,

c

is the constant of integration)

Options

A $x^{10}e^{5x} + c$
B $x^{10}e^{6x} + c$
C $x^{9}e^{5x} + c$
D $x^{10}e^{10x} + c$

Correct answer

B. $x^{10}e^{6x} + c$

Step-by-step solution

Using integration by parts in

{I}_{n},

we get,

{I}_{n} = ∫ \mathop{\mathop{x^{n}}_{⏟}}_{I}\mathop{\mathop{e^{6x}}_{⏟}}_{II}dx

= \left[\dfrac{x^{n}e^{6x}}{6}\right] - ∫ \dfrac{nx^{n - 1}e^{6x}}{6}dx

⇒ 6{I}_{n} = x^{n}e^{6x} - n{I}_{n - 1} + c

⇒ 6{I}_{n} + n{I}_{n - 1} = x^{n}e^{6x} + c

Put

n = 10

,

6{I}_{10} + 10{I}_{9} = x^{10}e^{6x} + c

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