match the integral to the solution\n∫(3x + 5) dx\n∫(x² + 5x - 1) dx\n∫(3) dx\n∫(3x² + 4x - 1) dx\n∫(-3x² + x…

Explanation:

Step1: Integrate $\int(3x + 5)dx$

Use the power - rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). $\int(3x + 5)dx=\int3x dx+\int5dx=3\times\frac{x^{1 + 1}}{1+1}+5x + c=\frac{3x^{2}}{2}+5x + c$

Step2: Integrate $\int(x^{2}+5x - 1)dx$

$\int(x^{2}+5x - 1)dx=\int x^{2}dx+\int5x dx-\int1dx=\frac{x^{3}}{3}+\frac{5x^{2}}{2}-x + c$

Step3: Integrate $\int(3)dx$

$\int(3)dx=3x + c$

Step4: Integrate $\int(3x^{2}+4x - 1)dx$

$\int(3x^{2}+4x - 1)dx=\int3x^{2}dx+\int4x dx-\int1dx=3\times\frac{x^{3}}{3}+4\times\frac{x^{2}}{2}-x + c=x^{3}+2x^{2}-x + c$

Step5: Integrate $\int(-3x^{2}+x - 1)dx$

$\int(-3x^{2}+x - 1)dx=-3\times\frac{x^{3}}{3}+\frac{x^{2}}{2}-x + c=-x^{3}+\frac{x^{2}}{2}-x + c$

Answer:

$\int(3x + 5)dx$ matches $\frac{3x^{2}}{2}+5x + c$ $\int(x^{2}+5x - 1)dx$ matches $\frac{x^{3}}{3}+\frac{5x^{2}}{2}-x + c$ $\int(3)dx$ matches $3x + c$ $\int(3x^{2}+4x - 1)dx$ matches $x^{3}+2x^{2}-x + c$ $\int(-3x^{2}+x - 1)dx$ matches $-x^{3}+\frac{x^{2}}{2}-x + c$