Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Find the integral
- Integrate by partial fractions
- Product of Binomials with Common Term
- FOIL Method
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Prove from LHS (left-hand side)
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Find the integral
Multiply the single term $2x+3$ by each term of the polynomial $\left(3x+5\right)$
Multiply the single term $3x$ by each term of the polynomial $\left(2x+3\right)$
When multiplying two powers that have the same base ($x$), you can add the exponents
Multiply the single term $5$ by each term of the polynomial $\left(2x+3\right)$
Combining like terms $9x$ and $10x$
Rewrite the integrand $\left(3x+5\right)\left(2x+3\right)$ in expanded form
Expand the integral $\int\left(6x^2+19x+15\right)dx$ into $3$ integrals using the sum rule for integrals, to then solve each integral separately
The integral of a function times a constant ($6$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$
Simplify the fraction $6\left(\frac{x^{3}}{3}\right)$
The integral $\int6x^2dx$ results in: $2x^{3}$
The integral of a function times a constant ($19$) is equal to the constant times the integral of the function
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$
Multiply the fraction and term in $19\cdot \left(\frac{1}{2}\right)x^2$
The integral $\int19xdx$ results in: $\frac{19}{2}x^2$
The integral of a constant is equal to the constant times the integral's variable
The integral $\int15dx$ results in: $15x$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
