We can solve the integral $\int\left(x-7\right)^{11}\left(x+3\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $x-7$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by finding the derivative of the equation above
Rewriting $x$ in terms of $u$
Substituting $u$, $dx$ and $x$ in the integral and simplify
Rewrite the integrand $u^{11}\left(u+10\right)$ in expanded form
Expand the integral $\int\left(u^{12}+10u^{11}\right)du$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
The integral $\int u^{12}du$ results in: $\frac{\left(x-7\right)^{13}}{13}$
The integral $\int10u^{11}du$ results in: $\frac{5}{6}\left(x-7\right)^{12}$
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 $C$
Try other ways to solve this exercise
Get a preview of step-by-step solutions.
Earn solution credits, which you can redeem for complete step-by-step solutions.
Save your favorite problems.
Become premium to access unlimited solutions, download solutions, discounts and more!
