বিভিন্ন সূত্রের ব্যবহারে যোগজীকরণ
∫cosx−1sinx+1;exdx\int { \cfrac { \cos { x } -1 }{ \sin { x } +1 } ; } { e }^{ x }dx∫sinx+1cosx−1;exdx is equal to:
excosx1+sinx+c\cfrac { { e }^{ x }\cos { x } }{ 1+\sin { x } } +c1+sinxexcosx+c
c−exsinx1+sinxc-\cfrac { { e }^{ x }\sin { x } }{ 1+\sin { x } } c−1+sinxexsinx
c−ex1+sinxc-\cfrac { { e }^{ x }}{ 1+\sin { x } } c−1+sinxex
c−excosx1+sinxc-\cfrac { { e }^{ x }\cos { x } }{ 1+\sin { x } } c−1+sinxexcosx
I=∫(cosx−1sinx+1)exdxI=∫ex(cosxsinx+1−11+sinx)dx Let f(x)=cosx1+sinxf′(x)=−sinx(1+sinx)−cos2x(1+sinx)2=−1−sinx(1+sinx)2=−11+sinxI=∫ex[f(x)+f′(x)]dxI=ex⋅f(x)=excosx1+sinxI=cosx⋅ex1+sinx+C \begin{array}{l} I=\int\left(\frac{\cos x-1}{\sin x+1}\right) e^{x} d x \\ I=\int e^{x}\left(\frac{\cos x}{\sin x+1}-\frac{1}{1+\sin x}\right) d x \\ \text { Let } f(x)=\frac{\cos x}{1+\sin x} \\ f^{\prime}(x)=\frac{-\sin x(1+\sin x)-\cos ^{2} x}{(1+\sin x)^{2}}=\frac{-1-\sin x}{(1+\sin x)^{2}}=-\frac{1}{1+\sin x} \\ I=\int e^{x}\left[f(x)+f^{\prime}(x)\right] d x \\ I=e^{x} \cdot f(x)=e^{x} \frac{\cos x}{1+\sin x} \\ I=\frac{\cos x \cdot e^{x}}{1+\sin x}+C \\ \end{array} I=∫(sinx+1cosx−1)exdxI=∫ex(sinx+1cosx−1+sinx1)dx Let f(x)=1+sinxcosxf′(x)=(1+sinx)2−sinx(1+sinx)−cos2x=(1+sinx)2−1−sinx=−1+sinx1I=∫ex[f(x)+f′(x)]dxI=ex⋅f(x)=ex1+sinxcosxI=1+sinxcosx⋅ex+C
∫sin(5−x10)dx=f(x)+c \int \sin{\left ( 5 - \frac{x}{10} \right )} dx = f{\left ( x \right )} + c ∫sin(5−10x)dx=f(x)+c হলে, f(x)এর মান কত?
g(x)=x g(x)=\sqrt{x} g(x)=x হলে-
i. ∫1g(x)dx=2x+c \int \frac{1}{g(x)} d x=2 \sqrt{x}+c ∫g(x)1dx=2x+c
ii. ∫01g(x)dx=23 \int_{0}^{1} g(x) d x=\frac{2}{3} ∫01g(x)dx=32
iii. ∫sec2xdxg(tanx)=2tanx+c \int \frac{\sec ^{2} x d x}{g(\tan x)}=2 \sqrt{\tan x}+c ∫g(tanx)sec2xdx=2tanx+c
নিচের কোনটি সঠিক?
∫dxx3 \int \frac{dx}{\sqrt[3]{x}} ∫3xdx সমান -
∫dxxx2−1=f(x)+c \int \frac{dx}{x \sqrt{x^{2} - 1}} = f{\left ( x \right )} + c ∫xx2−1dx=f(x)+c হলে f(x) এর মান-
