∫ inte^kx-এর সমাধান করুন

$$\int \int e ^ { k x }$$

মূল্যায়ন করুন

$$\left\{\begin{matrix}\frac{e^{kx}}{k^{2}}+Сx+С_{1},&k\neq 0\\Сx+\frac{x^{2}}{2}+С_{1},&k=0\end{matrix}\right.$$

সমাধানের ধাপগুলো

সিমপ্লিফাই।

$$\left(\frac{e^{kx}}{k}+С_{3}\right)x-\frac{-e^{kx}+kxe^{kx}}{k^{2}}$$

Integrate the sum term by term.

$$\int x\mathrm{d}x+\int С_{4}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}.

$$\frac{x^{2}}{2}+\int С_{4}\mathrm{d}x$$

Find the integral of С_{4} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{2}}{2}+С_{4}x$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\left\{\begin{matrix}\left(\frac{e^{kx}}{k}+С_{3}\right)x-\frac{-e^{kx}+kxe^{kx}}{k^{2}}+С_{7},&\\\frac{x^{2}}{2}+С_{4}x+С_{7},&\end{matrix}\right.$$

w.r.t. x পার্থক্য করুন

$$\left\{\begin{matrix}\frac{e^{kx}}{k}+С,&k\neq 0\\x+С,&k=0\end{matrix}\right.$$