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Chapter 1: Problem 2

Obtain a reduction formula for the following integrals (i) \(\int x^{n} e^{x} d x(n \geq 1)\) (ii) \(\int(\ln x)^{n} d x(n \geq 1)\)

Short Answer

Expert verified
Answer: The reduction formulas for the given integrals are: (i) \(I_n = x^{n} e^{x} - n I_{n-1}\) (ii) \(J_n = x (\ln x)^{n} - n J_{n-1}\)

Step by step solution

01

Integration by Parts

Integration by parts formula: \(\int u dv = uv - \int v du\). Choose \(u = x^{n}\) and \(dv = e^{x} dx\). Differentiate u and integrate dv: \(du = n x^{n-1} dx\) \(v = e^{x}\) Apply the formula: \(\int x^{n} e^{x} dx = x^{n} e^{x} - \int n x^{n-1} e^{x} dx\)
02

Reduction Formula

We can find the reduction formula by comparing the initial integral with the result we got: \(I_n = x^{n} e^{x} - n I_{n-1}\) where \(I_n = \int x^{n} e^{x} dx\) and \(I_{n-1} = \int x^{n-1} e^{x} dx\) For Integral (ii): \(\int(\ln x)^{n} dx\)
03

Integration by Parts

Integration by parts formula: \(\int u dv = uv - \int v du\). Choose \(u = (\ln x)^{n}\) and \(dv = dx\). Differentiate u and integrate dv: \(du = nx^{1-n} dx\) \(v = x\) Apply the formula: \(\int(\ln x)^{n} dx = x (\ln x)^{n} - n \int (\ln x)^{n-1} dx\)
04

Reduction Formula

We can find the reduction formula by comparing the initial integral with the result we got: \(J_n = x (\ln x)^{n} - n J_{n-1}\) where \(J_n = \int (\ln x)^{n} dx\) and \(J_{n-1} = \int (\ln x)^{n-1} dx\) In conclusion, the reduction formulas for the given integrals are: (i) \(I_n = x^{n} e^{x} - n I_{n-1}\) (ii) \(J_n = x (\ln x)^{n} - n J_{n-1}\)

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Most popular questions from this chapter

\(\int\left(x^{2}-2 x+3\right) \ell n x d x\)

Evaluate the following integrals : $$\int \frac{d x}{x \sqrt{\left(x^{2}-x+2\right)}}$$

Evaluate the following integrals: (i) \(\int \frac{\sqrt{2 x+1}}{x^{2}} d x\) (ii) \(\int \frac{x d x}{(a+b x)^{1 / 2}}\) (iii) \(\int \sqrt{\frac{x+a}{x+b}} d x\)

Evaluate the following integrals: (i) \(\int \frac{x}{(x-1)\left(x^{2}+4\right)} d x\) (ii) \(\int \frac{x^{3} d x}{x^{4}+3 x^{2}+2}\) (iii) \(\int \frac{x^{3}-1}{x^{3}+x} d\) (iv) \(\int \frac{x^{4}-2 x^{3}+3 x^{2}-x+3}{x^{3}-2 x^{2}+3 x} d x\)

Evaluate the following integrals: (i) \(\int \frac{2 x+\sin 2 x}{1+\cos 2 x} d x\) (ii) \(\int\left(\tan (\ln x)+\sec ^{2}(\ln x)\right\\} d x\) (iii) \(\int \frac{x+\sqrt{\left(1-x^{2}\right)} \sin ^{-1} x}{\sqrt{\left(1-x^{2}\right)}} d x\)
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