Evaluate the following indefinite integral by using the given substitution to reduce the integral to standard form
integral 2(2x+6)^5 dx, u=2x+6
you have
u^5 du
To evaluate the given indefinite integral using the given substitution, we need to first find the derivative of u with respect to x and then substitute the expression for u and du back into the integral.
Let's start by finding the derivative of u with respect to x:
du/dx = 2
Now, let's substitute the expression for u and du back into the integral:
integral 2(2x+6)^5 dx
= integral 2(u)^5 (1/2) du [using the substitution u = 2x+6 and du = 2dx]
Now we can simplify the integral:
= (1/2) integral 2(u)^5 du
= integral u^5 du
Now we can evaluate the integral of u^5 with respect to u:
= (1/6) u^6 + C [where C is the constant of integration]
Finally, let's substitute back the expression for u:
= (1/6) (2x+6)^6 + C
Therefore, the evaluated indefinite integral is (1/6) (2x+6)^6 + C.
To evaluate the given indefinite integral, we will use the substitution given \(u = 2x + 6\). Let's go through the steps:
Step 1: Calculate \(du\).
To find \(du\), we need to differentiate both sides of the equation \(u = 2x + 6\) with respect to \(x\). Since \(u\) is a function of \(x\), we can apply the Chain Rule. Differentiating \(2x + 6\) with respect to \(x\) gives us \(2\). Therefore, \(du = 2 \, dx\).
Step 2: Rewrite the integral in terms of \(u\) and \(du\).
Since \(du = 2 \, dx\), we can substitute \(du\) in place of \(2 \, dx\) in the integral:
\[\int 2(2x + 6)^5 \, dx = \int (2u)^5 \, \frac{du}{2} = 2^5 \cdot \frac{1}{2} \int u^5 \, du = 16 \int u^5 \, du\]
Step 3: Evaluate the indefinite integral of \(u^5 \, du\).
The integral of \(u^5 \, du\) can be evaluated using the power rule for integration. According to the power rule, when integrating \(x^n\) with respect to \(x\), we add 1 to the exponent and divide by the new exponent:
\[\int u^5 \, du = \frac{1}{6}u^{5+1} + C = \frac{1}{6}u^6 + C\]
where \(C\) represents the constant of integration.
Step 4: Substitute back in terms of \(x\).
Now, we substitute the value of \(u\) back in terms of \(x\), since the original integral was given in terms of \(x\):
\[\frac{1}{6}u^6 + C = \frac{1}{6}(2x + 6)^6 + C\]
Therefore, the evaluation of the indefinite integral \(\int 2(2x + 6)^5 \, dx\) using the given substitution \(u = 2x + 6\) is \(\frac{1}{6}(2x + 6)^6 + C\), where \(C\) is the constant of integration.