Integration by Substitution

Trending Questions

Q. $\int \frac{cosx+xsinx}{x(x+cosx)}dx=$

1. $-log|\frac{x}{x+cosx}|+c$
2. $log|\frac{x+cosx}{2x}|+c$
3. $-log|\frac{x+cosx}{2x}|+c$
4. $log|\frac{x}{x+cosx}|+c$

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Q. If $\int \left(\frac{x-1}{x+1}\right)\frac{dx}{\sqrt{x^3+x^2+x}}=2ta{n}^{-1}\sqrt{f\left(x\right)}+C$, find f(x).

1. $x+\frac{1}{x}+1$
2. $x+\frac{1}{x}+2$
3. $x-\frac{1}{x}+1$
4. $x-\frac{1}{x}-1$

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Q. $\int \frac{(x-x^5{)}^{1/5}}{x^6}dx$ is equal to

1. $\frac{5}{24}{(\frac{1}{x^4}-1)}^{6/5}+C$
2. $\frac{5}{24}{(1-\frac{1}{x^4})}^{6/5}+C$
3. $-\frac{5}{24}{(\frac{1}{x^4}-1)}^{6/5}+C$
4. None of these

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Q. $\int \frac{dx}{x^{\frac{1}{2}}(1+x^2{)}^{5/4}}$ is equal to

1. $\frac{2\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$
2. $\frac{-\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$
3. $\frac{\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$
4. $\frac{-2\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$

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Q. $\int \frac{(x+1)}{x(1+x{e}^x{)}^2}dx$ is equal to

1. $log\left(\frac{x}{1+x{e}^x}\right)+\frac{1}{1+x{e}^x}+C$
2. None of these
3. $log\left(\frac{1+x{e}^x}{x{e}^x}\right)+\frac{1}{1+x{e}^x}+C$
4. $log\left(\frac{x{e}^x}{1+x{e}^x}\right)+\frac{1}{1+e{x}^x}+C$

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Q. Let $S\left(x\right)=\int \frac{dx}{e^x+8e^{-x}+4e^{-3x}},R\left(x\right)=\int \frac{dx}{e^{3x}+8e^x+4e^{-x}}$ and $M\left(x\right)=S\left(x\right)-2R\left(x\right).$ If $M\left(x\right)=\frac{1}{2}ta{n}^{-1}\left(f\right(x\left)\right)+c$ then f(0)=

1. $\frac{3}{2}$
2. $\frac{1}{2}$
3. $\frac{5}{2}$
4. $\frac{7}{2}$

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Q. $\int \frac{x^3}{(1+x^2{)}^{1/3}}dx$ is equal to

1. $\frac{20}{3}(1+x^2{)}^{\frac{2}{3}}(2x^2-3)+C$
2. $\frac{3}{20}(1+x^2{)}^{\frac{2}{3}}(2x^2+3)+C$
3. None of these
4. $\frac{3}{20}(1+x^2{)}^{\frac{2}{3}}(2x^2-3)+C$

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Q. $\int x.(x^x{)}^x.(2logx+1)dx$ is equal to

1. $x^{\left(x^x\right)}+C$
2. $(x^x{)}^x+C$
3. $x^x.logx+C$
4. None of these

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Q. $\int \frac{\sqrt{5+x^{10}}}{x^{16}}dx=$

1. $\frac{-1}{75}(1+\frac{5}{x^{10}})+c$
2. $\frac{-1}{75}{(1+\frac{5}{x^{10}})}^{3/2}+c$
3. $\frac{-1}{50}{(1+\frac{5}{x^{10}})}^{3/2}+c$
4. $\frac{-1}{75}{(1+\frac{5}{x^{10}})}^{5/2}+c$

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Q. $\int \frac{\left[f\right(x\left)g^{{}^{\prime }}\right(x)+g(x\left)f^{{}^{\prime }}\right(x\left)\right]}{f\left(x\right).g\left(x\right)}\left[logf\right(x)+logg(x\left)\right]dx$ is equal to

1. $f\left(x\right)g\left(x\right)log\left(f\right(x\left)g\right(x)+C$
2. $\frac{1}{2}\left[logf\right(x\left)g\right(x){]}^2+C$
3. $\left[logf\right(x\left)g\right(x){]}^2+C$
4. $logf\left(x\right).g\left(x\right)+C$

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