MHT CET 2023 13th May Evening Shift | Indefinite Integration Question 10 | Mathematics

If $$\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}} d t=\frac{1}{2}[g(t)]^2+c$$, (where $$c$$ is a constant of integration), then $$g(2)$$ is

$$\frac{1}{\sqrt{5}} \log (2+\sqrt{5})$$

$$\frac{1}{2} \log (2+\sqrt{5})$$

$$2 \log (2+\sqrt{5})$$

$$\log (2+\sqrt{5})$$

MHT CET 2023 13th May Evening Shift

MCQ (Single Correct Answer)

-0

$$\int \frac{x-3}{(x-1)^3} e^x d x=$$

$$e^x\left(\frac{1}{(x-1)^2}\right)+c$$, where $$c$$ is constant of integration

$$e^x\left(\frac{1}{x+1}\right)+c$$, where $$c$$ is constant of integration

$$e^x\left((x-1)^2\right)+c$$, where $$c$$ is constant of integration

$$e^x\left((x-1)^3\right)+c$$, where $$c$$ is constant of integration

MHT CET 2023 13th May Evening Shift

MCQ (Single Correct Answer)

-0

$$\int \frac{2+\cos \frac{x}{2}}{x+\sin \frac{x}{2}} d x=$$

$$2 \log \left(x+\sin \frac{x}{2}\right)+c$$, where $$c$$ is constant of integration

$$\frac{1}{2} \log \left(x+\sin \frac{x}{2}\right)+c$$, where $$c$$ is constant of integration

$$4 \log \left(x+\sin \frac{x}{2}\right)+c$$, where $$c$$ is constant of integration

$$\log \left(x+\sin \frac{x}{2}\right)+c$$, where $$c$$ is constant of integration

MHT CET 2023 13th May Evening Shift

MCQ (Single Correct Answer)

-0

If $$I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x$$ and $$J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x$$, then for any arbitrary constant $$C$$, than the value of $$J-I$$ equals

$$\frac{1}{2} \log \left|\left(\frac{e^{4 x}-e^{2 x}+1}{e^{4 x}+e^{2 x}+1}\right)\right|+C$$

$$\frac{1}{2} \log \left|\left(\frac{e^{2 x}+e^x+1}{e^{2 x}-e^x+1}\right)\right|+C$$

$$\frac{1}{2} \log \left|\left(\frac{e^{2 x}-e^x+1}{e^{2 x}+e^x+1}\right)\right|+C$$

$$\frac{1}{2} \log \left|\left(\frac{e^{4 x}+e^{2 x}+1}{e^{4 x}-e^{2 x}+1}\right)\right|+C$$

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