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Yukti Vault

Calculus

Indefinite Integration: JEE Main & Advanced PYQs

125 previous year questions on Indefinite Integration from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 101-125 of 125

  1. Evaluate $\displaystyle\int\frac{(x+1)}{x\left(1+xe^{x}\right)^{2}}\,dx$.
  2. Find the indefinite integral $\displaystyle\int\left(\frac{1}{\sqrt[3]{x}+\sqrt[4]{x}}+\frac{\ln\left(1+\sqrt[6]{x}\right)}{\sqrt[3]{x}+\sqrt{x}}\right)dx$.
  3. For any natural number $m$, evaluate $\displaystyle\int\left(x^{3m}+x^{2m}+x^{m}\right)\left(2x^{2m}+3x^{m}+6\right)^{1/m}dx,\ x>0$.
  4. Evaluate $\displaystyle\int\sin^{-1}\left(\frac{2x+2}{\sqrt{4x^{2}+8x+13}}\right)dx$.
  5. Integrate $\displaystyle\int\dfrac{x^{3}+3x+2}{(x^{2}+1)^{2}(x+1)}\,dx$.
  6. $\displaystyle\int\dfrac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx$ is equal to
  7. If …
  8. Let $f(x)=\int\frac{(2-x^2)\cdot e^x}{\left(\sqrt{1+x}\right)(1-x)^{3/2}}dx$. If $f(0)=0$, then $f\left(\frac12\right)$ is equal to :
  9. Let $I(x)=\int\frac{3\,dx}{(4x+6)\left(\sqrt{4x^2+8x+3}\right)}$ and $I(0)=\frac{\sqrt3}{4}+20$. $I\left(\frac12\right)=\frac{a\sqrt2}{b}+c$, where $a,b,c\in\mathbf{N}$ …
  10. Let $f(t)=\int\left(\frac{1-\sin(\log_et)}{1-\cos(\log_et)}\right)dt$, for $t>1$. If $f\left(e^{\pi/2}\right)=-e^{\pi/2}$ and $f\left(e^{\pi/4}\right)=\alpha e^{\pi/4}$ …
  11. If $\int\left(\frac{1-5\cos^2x}{\sin^5x\cos^2x}\right)dx=f(x)+C$, where $C$ is the constant of integration, then $f\left(\frac\pi6\right)-f\left(\frac\pi4\right)$ is …
  12. Let $f(x)=\int\frac{dx}{x^{\left(\frac23\right)}+2x^{\left(\frac12\right)}}$ be such that $f(0)=-26+24\log_e(2)$. If $f(1)=\mathrm a+\mathrm b\log_e(3)$, where …
  13. Let $F(x)$ be an indefinite integral of $\sin^2 x$. STATEMENT-1 : The function $F(x)$ satisfies $F(x+\pi)=F(x)$ for all real $x$. because STATEMENT-2 …
  14. Let $f(x)=\frac{x}{(1+x^n)^{1/n}}$ for $n\geq2$ and $g(x)=(\underbrace{f\circ f\circ\cdots\circ f}_{f\text{ occurs }n\text{ times}})(x)$. Then $\int x^{n-2}g(x)\,dx$ …
  15. Let $$I=\int\frac{e^{x}}{e^{4x}+e^{2x}+1}\,dx,\quad J=\int\frac{e^{-x}}{e^{-4x}+e^{-2x}+1}\,dx.$$ Then, for an arbitrary constant $C$, the value of $J-I$ equals
  16. The integral $\displaystyle\int\frac{\sec^2 x}{(\sec x+\tan x)^{9/2}}\,dx$ equals (for some arbitrary constant $K$)
  17. Let $\int x^3\sin x\,dx=g(x)+C$, where $C$ is the constant of integration. If …
  18. If the integral $\int\frac{5\tan x}{\tan x-2}\,dx=x+a\ln|\sin x-2\cos x|+k$ then $a$ is equal to :
  19. If $\displaystyle\int\sin^{-1}\left(\dfrac{2x+2}{\sqrt{4x^{2}+8x+13}}\right)dx=A\tan^{-1}\left(\dfrac{2x+2}{3}\right)+B\log_{e}\left(4x^{2}+8x+13\right)+C$ ($C$ is a …
  20. If …
  21. If $\displaystyle\int\dfrac{\cos\theta}{5+7\sin\theta-2\cos^{2}\theta}\,d\theta=A\log_e|B(\theta)|+C$, where $C$ is a constant of integration, then …
  22. If $\displaystyle\int e^{x}\left(\dfrac{x\sin^{-1}x}{\sqrt{1-x^{2}}}+\dfrac{\sin^{-1}x}{\left(1-x^{2}\right)^{3/2}}+\dfrac{x}{1-x^{2}}\right)dx=g(x)+C$, where $C$ is the …
  23. Let $I(x)=\displaystyle\int\dfrac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If …
  24. If $\displaystyle\int\dfrac{2x^{2}+5x+9}{\sqrt{x^{2}+x+1}}\,dx=x\sqrt{x^{2}+x+1}+\alpha\sqrt{x^{2}+x+1}+\beta\log_e\left|x+\dfrac{1}{2}+\sqrt{x^{2}+x+1}\right|+C$, where …
  25. If $f(x)=\displaystyle\int\dfrac{1}{x^{1/4}\left(1+x^{1/4}\right)}\,dx$, $f(0)=-6$, then $f(1)$ is equal to :

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