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Problem 70

$$ \begin{aligned} &\text { If } y^{2}=a x^{2}+2 b x+c, \text { and } u_{n}=\int \frac{x^{n}}{y} d x, \text { prove that }\\\ &(n+1) a u_{n+1}+(2 n+1) b u_{n}+{ }^{n} c y_{n-1}=x^{n} y \text { and deduce }\\\ &\text { that } a u_{1}=y-b u_{0} ; 2 a^{2} u_{2}=y(a x-3 b)-\left(a c-3 b^{2}\right) u_{0} \text { . } \end{aligned} $$

Problem 71

\(\int \frac{\sec ^{2} x}{(\sec x+\tan x)^{9 / 2}} d x\) equals to (for some arbitrary constant \(K\) ) [Only One Correct Option 2012] (a) \(\frac{-1}{(\sec x+\tan x)^{1 / 2}}\left\\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^{2}\right\\}+K\) (b) \(\frac{1}{(\sec x+\tan x)^{1 / 2}}\left\\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^{2}\right\\}+K\) (c) \(\frac{-1}{(\sec x+\tan x)^{1 / 2}}\left\\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^{2}\right\\}+K\) (d) \(\frac{1}{(\sec x+\tan x)^{1 / 2}}\left\\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^{2}\right\\}+K\)

Problem 72

If \(I=\int \frac{e^{x}}{e^{4 x}+e^{2 x}+1} d x, \exists=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x .\) Then, for an arbitrary constant \(c\), the value of \(\exists-I\) equals [Only One Correct Option 2008] (a) \(\frac{1}{2} \log \left|\frac{e^{4 x}-e^{2 x}+1}{e^{4 x}+e^{2 x}+1}\right|+C\) (b) \(\frac{1}{2} \log \left|\frac{e^{2 x}+e^{x}+1}{e^{2 x}-e^{x}+1}\right|+C\) (c) \(\frac{1}{2} \log \left|\frac{e^{2 x}-e^{x}+1}{e^{2 x}+e^{x}+1}\right|+C\) (d) \(\frac{1}{2} \log \left|\frac{e^{4 x}+e^{2 x}+1}{e^{4 x}-e^{2 x}+1}\right|+C\)

Problem 73

The integral \(\int \frac{2 x^{12}+5 x^{9}}{\left(x^{5}+x^{3}+1\right)^{3}} d x\) is equal to (a) \(\frac{-x^{5}}{\left(x^{5}+x^{3}+1\right)^{2}}+C\) (b) \(\frac{x^{10}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C\) (c) \(\frac{x^{5}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C\) (d) \(\frac{-x^{10}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C\) where, \(C\) is an arbitrary constant.

Problem 75

The integral \(\int\left(1+x-\frac{1}{x}\right) e^{x+\frac{1}{x}} d x\) is equal to (a) \((x-1) e^{x+\frac{1}{x}}+C \quad\) (b) \(x e^{x+\frac{1}{x}}+C\) (c) \((x+1) e^{x+\frac{1}{x}}+C\) (d) \(-x e^{x+\frac{1}{x}}+C\)

Problem 76

If \(\int f(x) d x=\psi(x)\), then \(\int x^{5} f\left(x^{3}\right) d x\) is equal to [2013 JEE Main] (a) \(\frac{1}{3}\left[x^{3} \psi\left(x^{3}\right)-\int x^{2} \psi\left(x^{3}\right) d x\right]+C\) (b) \(\frac{1}{3} x^{3} \psi\left(x^{3}\right)-3 \int x^{3} \psi\left(x^{3}\right) d x+C\) (c) \(\frac{1}{3} x^{3} \psi\left(x^{3}\right)-\int x^{2} \psi\left(x^{3}\right) d x+C\) (d) \(\frac{1}{3}\left[x^{3} \psi\left(x^{3}\right)-\int x^{3} \psi\left(x^{3}\right) d x\right]+C\)

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