Name: Indefinite Integration by Substitution
Uploaded: 2019-07-19
Duration: 25 min 48 s
Description: We know the basic formulae to solve integration, but can we be satisfied with just these many formulae? Don't we need to find out the integration of more fun...

Question

Use integration by parts twice to find the exact value of each of the integral $\int_{0}^{\frac{1}{4} π} x^{2} \cos 2 x d x$

Accepted Answer

Let $I = \int_{0}^{\frac{π}{4}} x^{2} \cos 2 x d x$

Consider, $\int x^{2} \cos 2 x d x$

Let us perform integration by parts by taking $x^{2}$ as first function and $\cos 2 x$ as second function.

$∴ \int x^{2} \cos 2 x d x = x^{2} \int \cos 2 x d x - \int \{\frac{d}{d x} x^{2} \int \cos 2 x d x\} d x$

$= x^{2} \cdot \frac{\sin 2 x}{2} - \int (2 x \cdot \frac{\sin 2 x}{2}) d x$

$= \frac{x^{2} \sin 2 x}{2} - \int x \sin 2 x d x$

Again perform integration by parts in $\int x \sin 2 x d x$ by taking $x$ as first function and $\sin 2 x$ as second function.

$∴ \frac{x^{2} \sin 2 x}{2} - \int x \sin 2 x d x = \frac{x^{2} \sin 2 x}{2} - [x \int \sin 2 x d x - \int \{\frac{d}{d x} (x) \int \sin 2 x d x\} d x]$

$= \frac{x^{2} \sin 2 x}{2} - [\frac{- x \cos 2 x}{2} - \int (1 \times \frac{- \cos 2 x}{2}) d x]$

$= \frac{x^{2} \sin 2 x}{2} - [\frac{- x \cos 2 x}{2} + \int \frac{\cos 2 x}{2} d x]$

$= \frac{x^{2} \sin 2 x}{2} - [\frac{- x \cos 2 x}{2} + \frac{\sin 2 x}{4}] + C$

$= \frac{x^{2} \sin 2 x}{2} + \frac{x \cos 2 x}{2} - \frac{\sin 2 x}{4} + C$

Hence, $I = {[\frac{x^{2} \sin 2 x}{2} + \frac{x \cos 2 x}{2} - \frac{\sin 2 x}{4}]}_{0}^{\frac{π}{4}}$

$= \frac{\frac{π^{2}}{16} \sin (\frac{π}{2})}{2} + \frac{\frac{π}{4} \cos (\frac{π}{2})}{2} - \frac{\sin (\frac{π}{2})}{4} - 0 - 0 + \frac{\sin (0)}{4}$

$= \frac{π^{2}}{32} - \frac{1}{4}$

$= \frac{1}{32} (π^{2} - 8)$

Use integration by parts twice to find the exact value of each of the integral $\int_{0}^{\frac{1}{4} π} x^{2} \cos 2 x d x$

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Important Questions on Further Calculus

Learn and Prepare for any exam you want !

Use integration by parts twice to find the exact value of each of the integral $\int_{0}^{\frac{1}{4} π} x^{2} \cos 2 x d x$