Integration by Parts
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Integral Calculus

86233 \(\int \frac{x^{2}+1}{x^{4}+x^{2}+1} d x=\)

1 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{\sqrt{3}}\right)+c\)
2 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{3}\right)+c\)
3 \(\frac{1}{3} \tan ^{-1}\left(\frac{x+\frac{1}{x}}{3}\right)+c\)
4 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{\mathrm{x}+\frac{1}{\mathrm{x}}}{\sqrt{3}}\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86234 \(\int \frac{d x}{(x+2) \sqrt{x+1}}=\)

1 \(2 \tan ^{-1}(\sqrt{x+1})+c\)
2 \(\tan ^{-1}(\sqrt{x+1})+c\)
3 \(2 \tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
4 \(\tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86235 \(\int \mathrm{x}^{3} \cdot \mathrm{e}^{\mathrm{x}^{2}} \mathrm{dx}=\)

1 \(\frac{1}{2} \mathrm{ex}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
2 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
3 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
4 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86236 \(\int \frac{d x}{\sqrt{(x-1)(x-2)}}=\)

1 \(\log \left|\left(x-\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
2 \(\log \left|\left(\mathrm{x}-\frac{3}{2}\right)-\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
3 \(\log \left|\left(x+\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
4 \(\log \left|(\mathrm{x}-1)+\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86237 \(\int e^{x} \cdot \sec x(1+\tan x) d x=\)

1 \(e^{x} \sec x+c\)
2 \(\mathrm{e}^{\mathrm{x}} \tan \mathrm{x}+\mathrm{c}\)
3 \(\mathrm{e}^{\mathrm{x}} \cos \mathrm{ec} \mathrm{x}+\mathrm{c}\)
4 \(e^{x} \cot x+c\)
MHT CET-2020
NEET DIGITAL LIBRARY
Integral Calculus

86233 \(\int \frac{x^{2}+1}{x^{4}+x^{2}+1} d x=\)

1 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{\sqrt{3}}\right)+c\)
2 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{3}\right)+c\)
3 \(\frac{1}{3} \tan ^{-1}\left(\frac{x+\frac{1}{x}}{3}\right)+c\)
4 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{\mathrm{x}+\frac{1}{\mathrm{x}}}{\sqrt{3}}\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86234 \(\int \frac{d x}{(x+2) \sqrt{x+1}}=\)

1 \(2 \tan ^{-1}(\sqrt{x+1})+c\)
2 \(\tan ^{-1}(\sqrt{x+1})+c\)
3 \(2 \tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
4 \(\tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86235 \(\int \mathrm{x}^{3} \cdot \mathrm{e}^{\mathrm{x}^{2}} \mathrm{dx}=\)

1 \(\frac{1}{2} \mathrm{ex}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
2 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
3 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
4 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86236 \(\int \frac{d x}{\sqrt{(x-1)(x-2)}}=\)

1 \(\log \left|\left(x-\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
2 \(\log \left|\left(\mathrm{x}-\frac{3}{2}\right)-\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
3 \(\log \left|\left(x+\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
4 \(\log \left|(\mathrm{x}-1)+\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86237 \(\int e^{x} \cdot \sec x(1+\tan x) d x=\)

1 \(e^{x} \sec x+c\)
2 \(\mathrm{e}^{\mathrm{x}} \tan \mathrm{x}+\mathrm{c}\)
3 \(\mathrm{e}^{\mathrm{x}} \cos \mathrm{ec} \mathrm{x}+\mathrm{c}\)
4 \(e^{x} \cot x+c\)
MHT CET-2020
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Integral Calculus

86233 \(\int \frac{x^{2}+1}{x^{4}+x^{2}+1} d x=\)

1 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{\sqrt{3}}\right)+c\)
2 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{3}\right)+c\)
3 \(\frac{1}{3} \tan ^{-1}\left(\frac{x+\frac{1}{x}}{3}\right)+c\)
4 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{\mathrm{x}+\frac{1}{\mathrm{x}}}{\sqrt{3}}\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86234 \(\int \frac{d x}{(x+2) \sqrt{x+1}}=\)

1 \(2 \tan ^{-1}(\sqrt{x+1})+c\)
2 \(\tan ^{-1}(\sqrt{x+1})+c\)
3 \(2 \tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
4 \(\tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86235 \(\int \mathrm{x}^{3} \cdot \mathrm{e}^{\mathrm{x}^{2}} \mathrm{dx}=\)

1 \(\frac{1}{2} \mathrm{ex}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
2 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
3 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
4 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86236 \(\int \frac{d x}{\sqrt{(x-1)(x-2)}}=\)

1 \(\log \left|\left(x-\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
2 \(\log \left|\left(\mathrm{x}-\frac{3}{2}\right)-\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
3 \(\log \left|\left(x+\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
4 \(\log \left|(\mathrm{x}-1)+\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86237 \(\int e^{x} \cdot \sec x(1+\tan x) d x=\)

1 \(e^{x} \sec x+c\)
2 \(\mathrm{e}^{\mathrm{x}} \tan \mathrm{x}+\mathrm{c}\)
3 \(\mathrm{e}^{\mathrm{x}} \cos \mathrm{ec} \mathrm{x}+\mathrm{c}\)
4 \(e^{x} \cot x+c\)
MHT CET-2020
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Integral Calculus

86233 \(\int \frac{x^{2}+1}{x^{4}+x^{2}+1} d x=\)

1 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{\sqrt{3}}\right)+c\)
2 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{3}\right)+c\)
3 \(\frac{1}{3} \tan ^{-1}\left(\frac{x+\frac{1}{x}}{3}\right)+c\)
4 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{\mathrm{x}+\frac{1}{\mathrm{x}}}{\sqrt{3}}\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86234 \(\int \frac{d x}{(x+2) \sqrt{x+1}}=\)

1 \(2 \tan ^{-1}(\sqrt{x+1})+c\)
2 \(\tan ^{-1}(\sqrt{x+1})+c\)
3 \(2 \tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
4 \(\tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86235 \(\int \mathrm{x}^{3} \cdot \mathrm{e}^{\mathrm{x}^{2}} \mathrm{dx}=\)

1 \(\frac{1}{2} \mathrm{ex}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
2 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
3 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
4 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86236 \(\int \frac{d x}{\sqrt{(x-1)(x-2)}}=\)

1 \(\log \left|\left(x-\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
2 \(\log \left|\left(\mathrm{x}-\frac{3}{2}\right)-\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
3 \(\log \left|\left(x+\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
4 \(\log \left|(\mathrm{x}-1)+\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86237 \(\int e^{x} \cdot \sec x(1+\tan x) d x=\)

1 \(e^{x} \sec x+c\)
2 \(\mathrm{e}^{\mathrm{x}} \tan \mathrm{x}+\mathrm{c}\)
3 \(\mathrm{e}^{\mathrm{x}} \cos \mathrm{ec} \mathrm{x}+\mathrm{c}\)
4 \(e^{x} \cot x+c\)
MHT CET-2020
NEET DIGITAL LIBRARY
Integral Calculus

86233 \(\int \frac{x^{2}+1}{x^{4}+x^{2}+1} d x=\)

1 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{\sqrt{3}}\right)+c\)
2 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{x-\frac{1}{x}}{3}\right)+c\)
3 \(\frac{1}{3} \tan ^{-1}\left(\frac{x+\frac{1}{x}}{3}\right)+c\)
4 \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{\mathrm{x}+\frac{1}{\mathrm{x}}}{\sqrt{3}}\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86234 \(\int \frac{d x}{(x+2) \sqrt{x+1}}=\)

1 \(2 \tan ^{-1}(\sqrt{x+1})+c\)
2 \(\tan ^{-1}(\sqrt{x+1})+c\)
3 \(2 \tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
4 \(\tan ^{-1}(\sqrt{\mathrm{x}+2})+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86235 \(\int \mathrm{x}^{3} \cdot \mathrm{e}^{\mathrm{x}^{2}} \mathrm{dx}=\)

1 \(\frac{1}{2} \mathrm{ex}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
2 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}^{2}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
3 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}+1\right)+\mathrm{c}\)
4 \(\frac{1}{2} \mathrm{e}^{\mathrm{x}}\left(\mathrm{x}^{2}-1\right)+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86236 \(\int \frac{d x}{\sqrt{(x-1)(x-2)}}=\)

1 \(\log \left|\left(x-\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
2 \(\log \left|\left(\mathrm{x}-\frac{3}{2}\right)-\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
3 \(\log \left|\left(x+\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c\)
4 \(\log \left|(\mathrm{x}-1)+\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}\right|+\mathrm{c}\)
MHT CET-2020
Integral Calculus

86237 \(\int e^{x} \cdot \sec x(1+\tan x) d x=\)

1 \(e^{x} \sec x+c\)
2 \(\mathrm{e}^{\mathrm{x}} \tan \mathrm{x}+\mathrm{c}\)
3 \(\mathrm{e}^{\mathrm{x}} \cos \mathrm{ec} \mathrm{x}+\mathrm{c}\)
4 \(e^{x} \cot x+c\)
MHT CET-2020