1
MHT CET 2023 10th May Morning Shift
+2
-0

If $$\int \sqrt{\frac{x-7}{x-9}} d x=A \sqrt{x^2-16 x+63}+\log \left|(x-8)+\sqrt{x^2-16 x+63}\right|+c,$$

(where $$\mathrm{c}$$ is a constant of integration) then $$\mathrm{A}$$ is

A
$$-1$$
B
$$\frac{1}{2}$$
C
$$1$$
D
$$\frac{-1}{2}$$
2
MHT CET 2023 10th May Morning Shift
+2
-0

$$\int \frac{1}{7-6 x-x^2} d x=$$

A
$$\frac{1}{4} \log \left(\frac{7+x}{1-x}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\frac{1}{8} \log \left(\frac{7+x}{1-x}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
C
$$\frac{1}{16} \log \left(\frac{7+x}{1-x}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
D
$$\frac{1}{32} \log \left(\frac{7+x}{1-x}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
3
MHT CET 2023 10th May Morning Shift
+2
-0

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

A
$$\sqrt{2} \log \tan \left(x+\frac{\pi}{4}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\frac{1}{\sqrt{2}} \log \tan \left(\frac{x}{2}+\frac{\pi}{8}\right)+c$$, where c is a constant of integration.
C
$$\frac{1}{\sqrt{2}} \log \left(\frac{\tan \frac{x}{2}-\sqrt{2}+1}{\tan \frac{x}{2}+\sqrt{2}+1}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
D
$$-\frac{1}{\sqrt{2}} \log \left(\frac{\tan \frac{x}{2}-(\sqrt{2}+1)}{\tan \frac{x}{2}+\sqrt{2}-1}\right)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
4
MHT CET 2023 10th May Morning Shift
+2
-0

If $$\mathrm{I}=\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}$$, then $$\mathrm{I}$$ is

A
$$\left(\frac{x^4+1}{x}\right)^{\frac{1}{4}}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
B
$$\frac{\left(x^4-1\right)^{\frac{1}{4}}}{x}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
C
$$-\frac{\left(x^4+1\right)^{\frac{1}{4}}}{x}+c$$, where $$\mathrm{c}$$ is a constant of integration.
D
$$-\left(\frac{x^4+1}{x}\right)^{\frac{1}{4}}+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration.
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