∫ (5x+2)^3 wrt x=-এর সমাধান করুন

$$\int ( 5 x + 2 ) ^ { 3 } d x =$$

মূল্যায়ন করুন

$$\frac{\left(5x+2\right)^{4}}{20}+С$$

সমাধানের ধাপগুলো

\left(5x+2\right)^{3} প্রসারিত করতে বাইনোমিয়াল উপপাদ্য \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} ব্যবহার করুন৷

$$\int 125x^{3}+150x^{2}+60x+8\mathrm{d}x$$

Integrate the sum term by term.

$$\int 125x^{3}\mathrm{d}x+\int 150x^{2}\mathrm{d}x+\int 60x\mathrm{d}x+\int 8\mathrm{d}x$$

Factor out the constant in each of the terms.

$$125\int x^{3}\mathrm{d}x+150\int x^{2}\mathrm{d}x+60\int x\mathrm{d}x+\int 8\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. 125 কে \frac{x^{4}}{4} বার গুণ করুন।

$$\frac{125x^{4}}{4}+150\int x^{2}\mathrm{d}x+60\int x\mathrm{d}x+\int 8\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. 150 কে \frac{x^{3}}{3} বার গুণ করুন।

$$\frac{125x^{4}}{4}+50x^{3}+60\int x\mathrm{d}x+\int 8\mathrm{d}x$$

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}. 60 কে \frac{x^{2}}{2} বার গুণ করুন।

$$\frac{125x^{4}}{4}+50x^{3}+30x^{2}+\int 8\mathrm{d}x$$

Find the integral of 8 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{125x^{4}}{4}+50x^{3}+30x^{2}+8x$$

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.

$$\frac{125x^{4}}{4}+50x^{3}+30x^{2}+8x+С$$

w.r.t. x পার্থক্য করুন

$$\left(5x+2\right)^{3}$$