Question:
\(\begin{array}{l} I_n\left(x\right)=\int_0^x\frac{1}{\left(t^2+5\right)^n}dt, n=1, 2, 3,\cdots\end{array}\)
Then
\(\begin{array}{l} I_n\left(x\right)=\int_0^x\frac{1}{\left(t^2+5\right)^n}dt, n=1, 2, 3,\cdots\end{array}\)
Then
Updated On: Mar 17, 2026
- \(50I_6, - 9I_5 = xI'_5\)
- \(50I_6, - 11I_5 = xI'_5\)
- \(50I_6, - 9I_5 = I'_5\)
- \(50I_6, - 11I_5 = I'_5\)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
\(\begin{array}{l} I_n\left(x\right)=\displaystyle\int\limits_0^x\frac{1}{\left(t^2+5\right)^n}dt\end{array}\)
\(\begin{array}{l} =\displaystyle\int\limits_0^x\frac{1}{\underset{I}{\underbrace{\left(t^2+5\right)^n}}}\times\underset{II}{\underbrace{I}}dt\end{array}\)
\(\begin{array}{l} \left.=\frac{t}{\left(t^2+5\right)^n} \right|^x_0-\displaystyle\int\limits_0^x\frac{-2nt}{\left(t^2+5\right)^{n+1}}\times t\ dt\end{array}\)
\(\begin{array}{l} =\frac{x}{\left(x^2+5\right)^n}+\displaystyle\int\limits_0^x2n\left(\frac{t^2+5-5}{\left(t^2+5\right)^{n+1}}\right)dt\end{array}\)
\(\begin{array}{l} I_n\left(x\right)=\frac{x}{\left(x^2+5\right)^n}+2n\ I_n\left(x\right)-10n\ I_{n+1}\left(x\right) \end{array}\)
\(\begin{array}{l} 10n\ I_{n+1}\left(x\right)-\left(2n-1\right)\ I_n\left(x\right)=xI’_n\left(x\right) \end{array}\)
\(\begin{array}{l} \text{For}~n=5\\50I_6\left(x\right)-9I_5\left(x\right)=xI’_5\left(x\right)\end{array}\)
\(\begin{array}{l} \text{For}~n=5\\50I_6\left(x\right)-9I_5\left(x\right)=xI’_5\left(x\right)\end{array}\)
Was this answer helpful?
14
2
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_____
View More Questions
Top JEE Main Integration by Parts Questions
- Let \( f(x) = \int_0^{x^2 \frac{t^2 - 8t + 15}{e^t}} dt, \, x \in \mathbb{R} \). Then the numbers of local maximum and local minimum points of \( f \), respectively, are:
- If \( \int \left( x \sin^{-1} x + \sin^{-1} x (1 - x^2)^{3/2} + \frac{x}{1 - x^2} \right) dx = g(x) + C \), where C is the constant of integration, then \( g\left(\frac{1}{2}\right) \) equals:
- Let \[ \int x^3 \sin x \, dx = g(x) + C, \quad \text{where \( C \) is the constant of integration.} \] If \[ g\left( \frac{\pi}{2} \right) + g\left( \frac{\pi}{2} \right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \quad \alpha, \beta, \gamma \in {Z}, \] then \[ \alpha + \beta - \gamma \text{ equals:} \]
- The integral $ \int_{-1}^{\frac{3}{2}} \left( \pi^2 x \sin(\pi x) \right) dx $ is equal to:
- If $\int\limits_0^1\left(x^{21}+x^{14}+x^7\right)\left(2 x^{14}+3 x^7+6\right)^{1 / 7} d x=\frac{1}{l}(11)^{m / n}$ where $l, m, n \in N , m$ and $n$ are coprime then $l+m+n$ 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 ______.
View More Questions
Concepts Used:
Integration by Parts
Integration by Parts is a mode of integrating 2 functions, when they multiplied with each other. For two functions ‘u’ and ‘v’, the formula is as follows:
∫u v dx = u∫v dx −∫u' (∫v dx) dx
- u is the first function u(x)
- v is the second function v(x)
- u' is the derivative of the function u(x)
The first function ‘u’ is used in the following order (ILATE):
- 'I' : Inverse Trigonometric Functions
- ‘L’ : Logarithmic Functions
- ‘A’ : Algebraic Functions
- ‘T’ : Trigonometric Functions
- ‘E’ : Exponential Functions
The rule as a diagram:
