1
MHT CET 2022 11th August Evening Shift
+2
-0

$$\int \frac{3 x-2}{(x+1)(x-2)^2} \mathrm{~d} x=$$

(where $$C$$ is a constant of integration)

A
$$\frac{-5}{9} \log (x+1)+\frac{5}{9} \log (x-2)-\frac{4}{3} \times \frac{1}{(x-2)}+C$$
B
$$\frac{-5}{9} \log (x+1)+\frac{5}{9} \log (x-2)-\frac{1}{x-2}+C$$
C
$$\frac{1}{9} \log (x+1)+\frac{5}{9} \log (x-2)-\frac{4}{3} \times \frac{1}{(x-2)}+C$$
D
$$\frac{-5}{9} \log (x+1)+\frac{1}{9} \log (x-2)-\frac{1}{x-2}+C$$
2
MHT CET 2022 11th August Evening Shift
+2
-0

$$\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x=$$

(where $$C$$ is a constant of integration.)

A
$$x+\sin x+\sin 2 x+C$$
B
$$x+\sin x+\sin 2 x-C$$
C
$$x+2 \sin x+2 \sin 2 x+C$$
D
None of these
3
MHT CET 2022 11th August Evening Shift
+2
-0

$$\text { If } \int e^{x^2} \cdot x^3 \mathrm{~d} x=e^{x^2} \cdot[f(x)+C]$$ (where $$C$$ is a constant of integration.) and $$f(1)=0$$, then value of $$f(2)$$ will be

A
$$\frac{-3}{2}$$
B
$$\frac{-1}{2}$$
C
$$\frac{3}{2}$$
D
$$\frac{1}{2}$$
4
MHT CET 2021 24th September Evening Shift
+2
-0

$$\int e^x\left(\frac{1+\sin x}{1+\cos x}\right) d x=$$

A
$$e^x \tan \frac{x}{2}+c$$
B
$$e^x \cot \frac{x}{2}+c$$
C
$$e^x \cos \frac{x}{2}+c$$
D
$$e^x \sin \frac{x}{2}+c$$
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