Top 50 Most Repeated INTEGRALS PYQs | JEE MAINS
A curated collection of the most important questions from INTEGRALS, fully solved with step-by-step concepts to prepare for JEE MAINS.
A curated collection of the most important questions from INTEGRALS, fully solved with step-by-step concepts to prepare for JEE MAINS.
Integrate term‑wise: \int 3x^2 dx = x^3, \int -4x dx = -2x^2, \int 1 dx = x. The antiderivative is x^3 - 2x^2 + x. Substituting the limits gives (8 - ...
Read Full Step-by-Step Solution →Choose $u = x$ (algebraic) and $dv = e^{2x}dx$ (exponential). Then $du = dx$ and $v = \frac{1}{2}e^{2x}$. Applying $\int u\,dv = uv - \int v\,du$ give...
Read Full Step-by-Step Solution →Decompose: (3x+2)/(x(x+1)) = A/x + B/(x+1). Solve: A=2, B=1 → integral = 2ln|x| + ln|x+1| + C....
Read Full Step-by-Step Solution →Using King property $f(x) \to f(\frac{\pi}{2}-x)$, sum of integrals is $\int_0^{\pi/2} 1 dx$....
Read Full Step-by-Step Solution →After partial fraction decomposition, integral yields constant 3 on simplification....
Read Full Step-by-Step Solution →Reduction formula: $ I_n = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n}I_{n-2} $. Coefficient of $ I_{n-2} $ is $ \frac{n-1}{n} $....
Read Full Step-by-Step Solution →Use substitution: Let $ u = 2x $, so $ du = 2dx $, hence $ dx = \frac{du}{2} $. Then $ \int \sin(2x)~dx = \frac{1}{2} \int \sin u~du = -\frac{1}{2}\co...
Read Full Step-by-Step Solution →Odd function $\sin^3 x$ integrates to 0; $\cos x$ integrates to 2 over symmetric limits....
Read Full Step-by-Step Solution →Let \(u = x^{2}+x+1\); then \(du = (2x+1)dx\). The integral becomes \(\int_{u=1}^{u=3} \sqrt{u}\,du = \frac{2}{3}u^{3/2}\big|_{1}^{3} = \frac{2}{3}(3\...
Read Full Step-by-Step Solution →Antiderivative $=x^{3}+x^{2}$. Substituting limits: $(2^{3}+2^{2})-(0+0)=8+4=12$....
Read Full Step-by-Step Solution →Standard integral: $\int \frac{1}{x^2+a^2}dx = \frac{1}{a}\tan^{-1}(\frac{x}{a})+C$. Here $a=2$....
Read Full Step-by-Step Solution →Antiderivative of 2x+5 is x²+5x. Evaluating from 1 to 3 gives (9+15)−(1+5)=24−6=18....
Read Full Step-by-Step Solution →Integrate term‑wise: \int 3x^2 dx = x^3, \int -4x dx = -2x^2, \int 1 dx = x. The antiderivative is x^3 - 2x^2 + x. Substituting the limits gives (8 - ...
Read Full Step-by-Step Solution →Choose $u = x$ (algebraic) and $dv = e^{2x}dx$ (exponential). Then $du = dx$ and $v = \frac{1}{2}e^{2x}$. Applying $\int u\,dv = uv - \int v\,du$ give...
Read Full Step-by-Step Solution →Decompose: (3x+2)/(x(x+1)) = A/x + B/(x+1). Solve: A=2, B=1 → integral = 2ln|x| + ln|x+1| + C....
Read Full Step-by-Step Solution →Using King property $f(x) \to f(\frac{\pi}{2}-x)$, sum of integrals is $\int_0^{\pi/2} 1 dx$....
Read Full Step-by-Step Solution →After partial fraction decomposition, integral yields constant 3 on simplification....
Read Full Step-by-Step Solution →Reduction formula: $ I_n = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n}I_{n-2} $. Coefficient of $ I_{n-2} $ is $ \frac{n-1}{n} $....
Read Full Step-by-Step Solution →Use substitution: Let $ u = 2x $, so $ du = 2dx $, hence $ dx = \frac{du}{2} $. Then $ \int \sin(2x)~dx = \frac{1}{2} \int \sin u~du = -\frac{1}{2}\co...
Read Full Step-by-Step Solution →Odd function $\sin^3 x$ integrates to 0; $\cos x$ integrates to 2 over symmetric limits....
Read Full Step-by-Step Solution →Let \(u = x^{2}+x+1\); then \(du = (2x+1)dx\). The integral becomes \(\int_{u=1}^{u=3} \sqrt{u}\,du = \frac{2}{3}u^{3/2}\big|_{1}^{3} = \frac{2}{3}(3\...
Read Full Step-by-Step Solution →Antiderivative $=x^{3}+x^{2}$. Substituting limits: $(2^{3}+2^{2})-(0+0)=8+4=12$....
Read Full Step-by-Step Solution →Standard integral: $\int \frac{1}{x^2+a^2}dx = \frac{1}{a}\tan^{-1}(\frac{x}{a})+C$. Here $a=2$....
Read Full Step-by-Step Solution →Antiderivative of 2x+5 is x²+5x. Evaluating from 1 to 3 gives (9+15)−(1+5)=24−6=18....
Read Full Step-by-Step Solution →Integrate term‑wise: \int 3x^2 dx = x^3, \int -4x dx = -2x^2, \int 1 dx = x. The antiderivative is x^3 - 2x^2 + x. Substituting the limits gives (8 - ...
Read Full Step-by-Step Solution →Choose $u = x$ (algebraic) and $dv = e^{2x}dx$ (exponential). Then $du = dx$ and $v = \frac{1}{2}e^{2x}$. Applying $\int u\,dv = uv - \int v\,du$ give...
Read Full Step-by-Step Solution →Decompose: (3x+2)/(x(x+1)) = A/x + B/(x+1). Solve: A=2, B=1 → integral = 2ln|x| + ln|x+1| + C....
Read Full Step-by-Step Solution →Using King property $f(x) \to f(\frac{\pi}{2}-x)$, sum of integrals is $\int_0^{\pi/2} 1 dx$....
Read Full Step-by-Step Solution →After partial fraction decomposition, integral yields constant 3 on simplification....
Read Full Step-by-Step Solution →Reduction formula: $ I_n = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n}I_{n-2} $. Coefficient of $ I_{n-2} $ is $ \frac{n-1}{n} $....
Read Full Step-by-Step Solution →Use substitution: Let $ u = 2x $, so $ du = 2dx $, hence $ dx = \frac{du}{2} $. Then $ \int \sin(2x)~dx = \frac{1}{2} \int \sin u~du = -\frac{1}{2}\co...
Read Full Step-by-Step Solution →Odd function $\sin^3 x$ integrates to 0; $\cos x$ integrates to 2 over symmetric limits....
Read Full Step-by-Step Solution →Let \(u = x^{2}+x+1\); then \(du = (2x+1)dx\). The integral becomes \(\int_{u=1}^{u=3} \sqrt{u}\,du = \frac{2}{3}u^{3/2}\big|_{1}^{3} = \frac{2}{3}(3\...
Read Full Step-by-Step Solution →Antiderivative $=x^{3}+x^{2}$. Substituting limits: $(2^{3}+2^{2})-(0+0)=8+4=12$....
Read Full Step-by-Step Solution →Standard integral: $\int \frac{1}{x^2+a^2}dx = \frac{1}{a}\tan^{-1}(\frac{x}{a})+C$. Here $a=2$....
Read Full Step-by-Step Solution →Antiderivative of 2x+5 is x²+5x. Evaluating from 1 to 3 gives (9+15)−(1+5)=24−6=18....
Read Full Step-by-Step Solution →Integrate term‑wise: \int 3x^2 dx = x^3, \int -4x dx = -2x^2, \int 1 dx = x. The antiderivative is x^3 - 2x^2 + x. Substituting the limits gives (8 - ...
Read Full Step-by-Step Solution →Choose $u = x$ (algebraic) and $dv = e^{2x}dx$ (exponential). Then $du = dx$ and $v = \frac{1}{2}e^{2x}$. Applying $\int u\,dv = uv - \int v\,du$ give...
Read Full Step-by-Step Solution →Decompose: (3x+2)/(x(x+1)) = A/x + B/(x+1). Solve: A=2, B=1 → integral = 2ln|x| + ln|x+1| + C....
Read Full Step-by-Step Solution →Using King property $f(x) \to f(\frac{\pi}{2}-x)$, sum of integrals is $\int_0^{\pi/2} 1 dx$....
Read Full Step-by-Step Solution →After partial fraction decomposition, integral yields constant 3 on simplification....
Read Full Step-by-Step Solution →Reduction formula: $ I_n = -\frac{\sin^{n-1}x \cos x}{n} + \frac{n-1}{n}I_{n-2} $. Coefficient of $ I_{n-2} $ is $ \frac{n-1}{n} $....
Read Full Step-by-Step Solution →Use substitution: Let $ u = 2x $, so $ du = 2dx $, hence $ dx = \frac{du}{2} $. Then $ \int \sin(2x)~dx = \frac{1}{2} \int \sin u~du = -\frac{1}{2}\co...
Read Full Step-by-Step Solution →Odd function $\sin^3 x$ integrates to 0; $\cos x$ integrates to 2 over symmetric limits....
Read Full Step-by-Step Solution →Let \(u = x^{2}+x+1\); then \(du = (2x+1)dx\). The integral becomes \(\int_{u=1}^{u=3} \sqrt{u}\,du = \frac{2}{3}u^{3/2}\big|_{1}^{3} = \frac{2}{3}(3\...
Read Full Step-by-Step Solution →Antiderivative $=x^{3}+x^{2}$. Substituting limits: $(2^{3}+2^{2})-(0+0)=8+4=12$....
Read Full Step-by-Step Solution →