Integration By Parts (Part-1)| S N Dey| Class 12

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In this article, we will discuss the solutions of VSA type Questions of Integration by Parts as given in the Chhaya Mathematics Class 12 written by S N Dey.

Integration by parts, S N Dey mathematics class 12
VSA Type Questions and Solutions ( Integration By Parts) :

Integrate :

1.~~\displaystyle\int{\log x~dx} \\=\log x\int dx-\displaystyle\int{\left[\frac{d}{dx}(\log x) \int{dx}\right]~dx}\\=x \log x-\displaystyle\int{\frac 1x \times x}~dx\\=x\log x-\int{dx}+c\\=x\log x-x+c

[c \rightarrow \text{constant of Integration}]

2.~~\displaystyle \int{x \cos x}~dx

Solution.

\displaystyle\int{x \cos x}~dx\\=x\int{\cos x}-\displaystyle\int{\left[\frac{d}{dx}(x) \int{\cos x~dx}\right]~dx}\\=x\sin x-\int{\sin x}~dx\\=x\sin x+\cos x+c.

3.~\displaystyle\int{xe^x}~dx

Solution.

\displaystyle\int{xe^x}~dx\\=x\displaystyle\int{e^x}~dx-\int{\left[\frac{d}{dx}(x) \int{e^x}~dx\right]}~dx\\=xe^x-\int{e^x}~dx\\=xe^x-e^x+c\\=e^x(x-1)+c.

4.~\displaystyle\int{x\sin x}~dx

Solution.

\displaystyle\int{x\sin x}~dx\\=x\displaystyle\int{\sin x}~dx-\int{\left[\frac{d}{dx}(x) \int{\sin x}~dx\right]}~dx\\=-x\cos x+\int{\cos x}~dx\\=-x\cos x+\sin x+c\\=\sin x-x\cos x+c.

5.~\displaystyle\int{x\sec^2 x}~dx

Solution.

\displaystyle\int{x\sec^2 x}~dx\\=x\displaystyle\int{\sec^2 x}~dx-\int{\left[\frac{d}{dx}(x) \int{\sec^2 x}~dx\right]}~dx\\=x\tan x-\int{\tan x}~dx\\=x\tan x-\log|\sec x|+c\\=x\tan x+\log|\cos x|+c.

7.~~\displaystyle\int{\cot^{-1} x}~dx

Solution.

~~\displaystyle\int{\cot^{-1} x}~dx\\=\cot^{-1}x \displaystyle\int{dx}-\displaystyle\int\left[\frac{d}{dx}(\cot^{-1} x) \int{dx}\right]~dx\\=x\cot^{-1}x+\int{\frac{1}{1+x^2}}\cdot x~dx\\=x\cot^{-1}x+\frac 12\int{\frac{d(1+x^2)}{1+x^2}}\\=x\cot^{-1}x+\frac 12\log|1+x^2|+c.

8.~~\displaystyle\int{\sec^{-1}x}~dx

Solution.

\displaystyle\int{\sec^{-1}x}~dx\\=\sec^{-1}x\int{dx}-\int{\left[\frac{d}{dx}(\sec^{-1}x)\int{dx}\right]}~dx\\=x\sec^{-1}x-\int{\frac{x}{x\sqrt{x^2-1}}}~dx\\=x\sec^{-1}x-\int{\frac{x}{\sqrt{x^2-1}}}~dx\\=x\sec^{-1}x-\log|x+\sqrt{x^2-1}|+c.

9.~~\displaystyle\int{\csc^{-1}x}~dx

Solution.

\displaystyle\int{\csc^{-1}x}~dx\\=\csc^{-1}x\int{dx}-\int{\left[\frac{d}{dx}(\csc^{-1}x)\int{dx}\right]~dx}\\=x\csc^{-1}x-\int{\left(-\frac{x}{x\sqrt{x^2-1}}\right)~dx}\\=x\csc^{-1}x+\int{\frac{dx}{\sqrt{x^2-1}}}\\=x\csc^{-1}x+\log|x+\sqrt{x^2-1}|+c.

10.~~\displaystyle\int{\cos^{-1}\frac 1x}~dx

Solution.

\displaystyle\int{\cos^{-1}\frac 1x}~dx\\=\int{\sec^{-1}x}\\=\sec^{-1}x\int{dx}-\int{\left[\frac{d}{dx}(\sec^{-1}x)\int{dx}\right]~dx}\\=x\sec^{-1}x-\int{\frac{x}{x\sqrt{x^2-1}}}~dx\\=x\sec^{-1}x-\int{\frac{dx}{\sqrt{x^2-1}}}\\=x\sec^{-1}x-\log|x+\sqrt{x^2-1}|+c.

11.~~\displaystyle\int{x^n \log x}~dx

Solution.

\displaystyle\int{x^n\log x}~dx\\=\log x \int{x^n}~dx-\int{\left[\frac{d}{dx}(\log x) \int{x^n}~dx\right]}~dx\\=\frac{x^{n+1}}{n+1} \log x-\int{\frac 1x\times \frac{x^{n+1}}{n+1}}~dx\\=\frac{x^{n+1}}{n+1} \log x-\frac{1}{n+1}\int{x^n}~dx\\=\frac{x^{n+1}}{n+1} \log x-\frac{1}{n+1} \cdot \frac{x^{n+1}}{n+1}\\=\frac{x^{n+1}}{n+1}\left(\log x-\frac{1}{n+1}\right)+c.

12.~~\displaystyle\int{x\sin^{-1}x}~dx

Solution.

\displaystyle\int{x\sin^{-1}x}~dx\\=\sin^{-1}x \int{x}~dx-\int{\left[\frac{d}{dx}(\sin^{-1}x)\int{x}~dx\right]}~dx\\=\frac{x^2}{2}\sin^{-1}x-\int{\frac{1}{\sqrt{1-x^2}}\times \frac{x^2}{2}}~dx\\=\frac{x^2}{2}\sin^{-1}x+\frac 12\int{\frac{1-x^2-1}{\sqrt{1-x^2}}}~dx\\=\frac{x^2}{2}\sin^{-1}x+\frac 12\int{\sqrt{1-x^2}}~dx-\frac 12\int{\frac{dx}{\sqrt{1-x^2}}}\\=\frac{x^2}{2}\sin^{-1}x+\frac 12\left(\frac{x\sqrt{1-x^2}}{2}+\frac 12\sin^{-1}x\right)-\frac 12 \sin^{-1}x+c\\=\frac{x^2}{2}\sin^{-1}x+\frac 14x\sqrt{1-x^2}-\frac 14\sin^{-1}x+c.

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