Let I(x) = \(∫√\frac{x+7}{x} dx\) and I (9) = 12 + 7 loge 7. If I (1) = α + 7 loge (1 + 2√2), then α4 is equal to_____.
Correct Answer: 64
Solution and Explanation
The given integral is:
\( \int \sqrt{\frac{x+7}{x}} \, dx \)
Step 1: Substitution
Let \( x = t^2 \), so:
\( dx = 2t \, dt \)
Substitute into the integral:
\( \int \sqrt{\frac{x+7}{x}} \, dx = 2 \int \sqrt{\frac{t^2+7}{t^2}} t \, dt \)
Simplify:
\( 2 \int \sqrt{t^2 + 7} \, dt = 2 \int (t + \sqrt{7}) \, dt \)
Integrate:
\( I(t) = 2 \left[ \frac{t}{2} \sqrt{t^2 + 7} + \frac{7}{2} \ln |t + \sqrt{t^2 + 7}| \right] + C \)
Simplify:
\( I(t) = t\sqrt{t^2 + 7} + 7 \ln |t + \sqrt{t^2 + 7}| + C \)
Substitute \( t = \sqrt{x} \) back:
\( I(x) = \sqrt{x} \sqrt{x+7} + 7 \ln |\sqrt{x} + \sqrt{x+7}| + C \)
Step 2: Finding the constant \( C \)
Given \( I(9) = 12 + 7 \ln 7 \):
\( I(9) = \sqrt{9} \sqrt{9+7} + 7 \ln |\sqrt{9} + \sqrt{9+7}| + C \)
Simplify:
\( 12 + 7 \ln 7 = 12 + 7 \ln (3+4) + C \)
\( C = 0 \)
Thus:
\( I(x) = \sqrt{x} \sqrt{x+7} + 7 \ln |\sqrt{x} + \sqrt{x+7}| \)
Step 3: Finding \( I(1) \) and \( \alpha \)
Substitute \( x=1 \):
\( I(1) = \sqrt{1} \sqrt{1+7} + 7 \ln |\sqrt{1} + \sqrt{1+7}| \)
\( I(1) = \sqrt{8} + 7\ln(1 + \sqrt{8}) \)
Let \( \alpha = \sqrt{8} \), so:
\( \alpha^4 = (\sqrt{8})^4 = 8^2 = 64 \)
Final Answer:
\( \alpha^4 = 64 \)
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main types of differential equations Questions
- Let the system of equations \(x+2y+3z = 5\), \(2x+3y+z = 9\), \(4x+3y+λz = μ\) have an infinite number of solutions. Then \(λ + 2μ\) is equal to
- Let \(y=y(t)\) be a solution of the differential equation\(\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}\) where, \(\alpha > 0, \beta>0\) and \(\gamma > 0\). Then \(\displaystyle\lim _{t \rightarrow \infty} y(t)\)
The area enclosed by the closed curve $C$ given by the differential equation $\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0$ is $4 \pi$.
Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $R S$ is
The statement
\((p⇒q)∨(p⇒r) \)
is NOT equivalent toLet α, β(α > β) be the roots of the quadratic equation x2 – x – 4 = 0.
If \(P_n=α^n–β^n, n∈N\) then \(\frac{P_{15}P_{16}–P_{14}P_{16}–P_{15}^2+P_{14}P_{15}}{P_{13}P_{14}}\)
is equal to _______.
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.