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1
MHT CET 2023 13th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

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

A
$$e^x\left(\frac{1}{(x-1)^2}\right)+c$$, where $$c$$ is constant of integration
B
$$e^x\left(\frac{1}{x+1}\right)+c$$, where $$c$$ is constant of integration
C
$$e^x\left((x-1)^2\right)+c$$, where $$c$$ is constant of integration
D
$$e^x\left((x-1)^3\right)+c$$, where $$c$$ is constant of integration
2
MHT CET 2023 13th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

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

A
$$2 \log \left(x+\sin \frac{x}{2}\right)+c$$, where $$c$$ is constant of integration
B
$$\frac{1}{2} \log \left(x+\sin \frac{x}{2}\right)+c$$, where $$c$$ is constant of integration
C
$$4 \log \left(x+\sin \frac{x}{2}\right)+c$$, where $$c$$ is constant of integration
D
$$\log \left(x+\sin \frac{x}{2}\right)+c$$, where $$c$$ is constant of integration
3
MHT CET 2023 13th May Evening Shift
MCQ (Single Correct Answer)
+2
-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

A
$$\frac{1}{2} \log \left|\left(\frac{e^{4 x}-e^{2 x}+1}{e^{4 x}+e^{2 x}+1}\right)\right|+C$$
B
$$\frac{1}{2} \log \left|\left(\frac{e^{2 x}+e^x+1}{e^{2 x}-e^x+1}\right)\right|+C$$
C
$$\frac{1}{2} \log \left|\left(\frac{e^{2 x}-e^x+1}{e^{2 x}+e^x+1}\right)\right|+C$$
D
$$\frac{1}{2} \log \left|\left(\frac{e^{4 x}+e^{2 x}+1}{e^{4 x}-e^{2 x}+1}\right)\right|+C$$
4
MHT CET 2023 13th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

If $$\mathrm{I}=\int \frac{2 x-7}{\sqrt{3 x-2}} \mathrm{~d} x$$, then $$\mathrm{I}$$ is given by

A
$$\frac{106}{27}(3 x-2)^{\frac{3}{2}}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\frac{98}{27}(3 x-2)^{\frac{3}{2}}+\mathrm{c}$$, where c is a constant of integration.
C
$$\frac{4}{27}(3 x-2)^{\frac{3}{2}}-\frac{34}{9}(3 x-2)^{\frac{1}{2}}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
D
$$\frac{4}{27}(3 x-2)^{\frac{3}{2}}+\frac{34}{9}(3 x-2)^{\frac{1}{2}}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration
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