ISC Class 12 Integrals — practice questions
74 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice ISC Class 12 Integrals in the app →Which function is an antiderivative of $\cos x$?The integral $\int x^n\,dx$ (for $n\neq -1$) equals$\int \dfrac{1}{x}\,dx$ equals$\int a^{x}\,dx$ (with $a>0,\ a\neq 1$) equals$\int ec^2 x\,dx$ equals$\int \dfrac{x^3-1}{x^2}\,dx$ equals$\int \left(x^{3/2}+2e^{x}-\dfrac{1}{x}\right)dx$ equals$\int \operatorname{cosec} x\,(\operatorname{cosec} x+\cot x)\,dx$ equals$\int \dfrac{1- in x}{\cos^2 x}\,dx$ equalsIf $F$ is the antiderivative of $f(x)=4x^3-6$ with $F(0)=3$, then $F(x)$ isThe antiderivative of $\left( qrt{x}+\dfrac{1}{ qrt{x}}\right)$ equalsUsing inspection, an antiderivative of $\cos 2x$ is$\int 2x\, in(x^2+1)\,dx$ equals$\int in mx\,dx$ equals$\int \tan x\,dx$ equals$\int \cot x\,dx$ equals$\int ec x\,dx$ equals$\int \dfrac{2x}{1+x^2}\,dx$ equals$\int \dfrac{(\log x)^2}{x}\,dx$ equals$\int \dfrac{e^{\tan^{-1}x}}{1+x^2}\,dx$ equals$\int \dfrac{10x^9+10^{x}\log_e 10}{x^{10}+10^{x}}\,dx$ equals$\int \dfrac{dx}{ in^2 x\cos^2 x}$ equals$\int \cos^2 x\,dx$ equalsUsing the identity $ in x\cos y=\tfrac12[ in(x+y)+ in(x-y)]$, the value of $\int in 2x\cos 3x\,dx$ is$\int in^3 x\,dx$ equals$\int \dfrac{dx}{x^2+a^2}$ equals$\int \dfrac{dx}{x^2-a^2}$ equals$\int \dfrac{dx}{ qrt{a^2-x^2}}$ equals$\int \dfrac{dx}{x^2-16}$ equals$\int \dfrac{dx}{ qrt{2x-x^2}}$ equals$\int \dfrac{dx}{x^2-6x+13}$ equals$\int \dfrac{dx}{x^2+2x+2}$ equalsTo integrate $\int \dfrac{px+q}{ax^2+bx+c}\,dx$, one writes $px+q=A\dfrac{d}{dx}(ax^2+bx+c)+B$. This equals$\int \dfrac{dx}{(x+1)(x+2)}$ equalsWhen decomposing $\dfrac{px^2+qx+r}{(x-a)^2(x-b)}$ into partial fractions, the correct form isIn decomposing $\dfrac{1}{(x+1)(x+2)}=\dfrac{A}{x+1}+\dfrac{B}{x+2}$, the constants are$\int \dfrac{x\,dx}{(x-1)(x-2)}$ equals$\int \dfrac{dx}{x(x^2+1)}$ equalsThe integration-by-parts formula $\int u\,\dfrac{dv}{dx}\,dx$ equals$\int x\cos x\,dx$ equals$\int \log x\,dx$ equals$\int x e^{x}\,dx$ equals$\int e^{x} in x\,dx$ equals$\int e^{x}\big[f(x)+f'(x)\big]\,dx$ equals$\int e^{x}\left(\tan^{-1}x+\dfrac{1}{1+x^2}\right)dx$ equals$\int x^{2}e^{x^{3}}\,dx$ equals$\int e^{x} ec x\,(1+\tan x)\,dx$ equals$\int qrt{x^2-a^2}\,dx$ equals$\int qrt{a^2-x^2}\,dx$ equals$\int qrt{x^2+2x+5}\,dx$ equals$\int qrt{3-2x-x^2}\,dx$ equalsBy the Second Fundamental Theorem of Calculus, if $F'=f$ on $[a,b]$, then $\int_a^b f(x)\,dx$ equals$\displaystyle\int_2^3 x^2\,dx$ equals$\displaystyle\int_1^{ qrt{3}} \dfrac{dx}{1+x^2}$ equals$\displaystyle\int_0^{2/3} \dfrac{dx}{4+9x^2}$ equals$\displaystyle\int_{-1}^{1} 5x^4 qrt{x^5+1}\,dx$ equals$\displaystyle\int_0^{1} \dfrac{\tan^{-1}x}{1+x^2}\,dx$ equalsIf $f(x)=\displaystyle\int_0^{x} t in t\,dt$, then $f'(x)$ isBy property $\mathbf{P_3}$ of definite integrals, $\displaystyle\int_a^b f(x)\,dx$ equalsIf $f$ is an odd function, then $\displaystyle\int_{-a}^{a} f(x)\,dx$ equals$\displaystyle\int_{-1}^{1} in^5 x\cos^4 x\,dx$ equals$\displaystyle\int_{-\pi/4}^{\pi/4} in^2 x\,dx$ equals$\displaystyle\int_0^{\pi/2} \dfrac{ in^4 x}{ in^4 x+\cos^4 x}\,dx$ equals$\displaystyle\int_{\pi/6}^{\pi/3} \dfrac{dx}{1+ qrt{\tan x}}$ equals$\displaystyle\int_0^{\pi/2} \log in x\,dx$ equalsUsing property $\mathbf{P_2}$, $\displaystyle\int_{-1}^{2} |x^3-x|\,dx$ equalsThe value of $\displaystyle\int_0^{\pi/2} \log\!\left(\dfrac{4+3 in x}{4+3\cos x}\right)dx$ is$\displaystyle\int_0^{1} x e^{x^2}\,dx$ equals$\displaystyle\int_0^{\pi/4} \tan x\,dx$ equalsA rational function $\dfrac{P(x)}{Q(x)}$ is called proper when$\displaystyle\int \frac{dx}{e^{x}+e^{-x}}$ is equal toIf $f(a+b-x)=f(x)$, then $\displaystyle\int_a^b x\,f(x)\,dx$ is equal to$\displaystyle\int_0^{2a} f(x)\,dx$ equals $2\displaystyle\int_0^{a} f(x)\,dx$ precisely when$\displaystyle\int_2^3 \dfrac{x\,dx}{x^2+1}$ equals