Question Statement

Evaluate the integral:

$\int (2 x + 7) (x^{2} + 7 x + 3)^{\frac{4}{5}} d x$

Background and Explanation

This problem is solved using integration by substitution (also called u-substitution). The key insight is recognizing that the derivative of the inner function $(x^{2} + 7 x + 3)$ is $(2 x + 7)$ , which appears as a separate factor in the integrand—this makes substitution straightforward.

Solution

Let $I$ denote the integral we need to evaluate:

$I = \int (2 x + 7) (x^{2} + 7 x + 3)^{\frac{4}{5}} d x$

We can rewrite this to group the terms strategically:

$I = \int (x^{2} + 7 x + 3)^{4/5} \cdot (2 x + 7) d x$

Step 1: Choose the substitution

Notice that if we let $t = x^{2} + 7 x + 3$ , then the derivative is:

$\frac{d t}{d x} = 2 x + 7$

Step 2: Find the differential

Multiplying both sides by $d x$ :

$(2 x + 7) d x = d t$

Step 3: Substitute into the integral

Replacing $(x^{2} + 7 x + 3)$ with $t$ and $(2 x + 7) d x$ with $d t$ in equation (1):

$I = \int (t)^{4/5} \cdot d t$

Step 4: Integrate using the power rule

Applying the power rule for integration $\int t^{n} d t = \frac{t ^{n + 1}}{n + 1} + C$ where $n = \frac{4}{5}$ :

$I = \frac{t ^{\frac{4}{5} + 1}}{\frac{4}{5} + 1} + c = \frac{t ^{\frac{9}{5}}}{\frac{9}{5}} + c = \frac{5}{9} t^{\frac{9}{5}} + c$

Step 5: Substitute back to the original variable

Replace $t$ with $(x^{2} + 7 x + 3)$ to get the final answer:

$I = \frac{5}{9} (x^{2} + 7 x + 3)^{\frac{9}{5}} + c$

Key Formulas or Methods Used

Integration by Substitution: If $u = g (x)$ , then $\int f (g (x)) g^{'} (x) d x = \int f (u) d u$
Power Rule for Integration: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ (for $n \neq = - 1$ )
Differential Calculation: $d (t) = \frac{d t}{d x} \cdot d x$

Summary of Steps

Identify the substitution: Let $t = x^{2} + 7 x + 3$ (the expression inside the power)
Calculate the differential: $d t = (2 x + 7) d x$ , which matches the remaining factor in the integrand
Rewrite the integral: Substitute to get $\int t^{4/5} d t$
Apply the power rule: Integrate to obtain $\frac{5}{9} t^{9/5} + C$
Back-substitute: Replace $t$ with $(x^{2} + 7 x + 3)$ to express the answer in terms of $x$

Question Statement

Evaluate the integral:

$\int (2 x + 7) (x^{2} + 7 x + 3)^{\frac{4}{5}} d x$

Background and Explanation

Solution

Let $I$ denote the integral we need to evaluate:

$I = \int (2 x + 7) (x^{2} + 7 x + 3)^{\frac{4}{5}} d x$

We can rewrite this to group the terms strategically:

$I = \int (x^{2} + 7 x + 3)^{4/5} \cdot (2 x + 7) d x$

Step 1: Choose the substitution

Notice that if we let $t = x^{2} + 7 x + 3$ , then the derivative is:

$\frac{d t}{d x} = 2 x + 7$

Step 2: Find the differential

Multiplying both sides by $d x$ :

$(2 x + 7) d x = d t$

Step 3: Substitute into the integral

Replacing $(x^{2} + 7 x + 3)$ with $t$ and $(2 x + 7) d x$ with $d t$ in equation (1):

$I = \int (t)^{4/5} \cdot d t$

Step 4: Integrate using the power rule

Applying the power rule for integration $\int t^{n} d t = \frac{t ^{n + 1}}{n + 1} + C$ where $n = \frac{4}{5}$ :

$I = \frac{t ^{\frac{4}{5} + 1}}{\frac{4}{5} + 1} + c = \frac{t ^{\frac{9}{5}}}{\frac{9}{5}} + c = \frac{5}{9} t^{\frac{9}{5}} + c$

Step 5: Substitute back to the original variable

Replace $t$ with $(x^{2} + 7 x + 3)$ to get the final answer:

$I = \frac{5}{9} (x^{2} + 7 x + 3)^{\frac{9}{5}} + c$

Key Formulas or Methods Used

Integration by Substitution: If $u = g (x)$ , then $\int f (g (x)) g^{'} (x) d x = \int f (u) d u$
Power Rule for Integration: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ (for $n \neq = - 1$ )
Differential Calculation: $d (t) = \frac{d t}{d x} \cdot d x$

Summary of Steps

Identify the substitution: Let $t = x^{2} + 7 x + 3$ (the expression inside the power)
Calculate the differential: $d t = (2 x + 7) d x$ , which matches the remaining factor in the integrand
Rewrite the integral: Substitute to get $\int t^{4/5} d t$
Apply the power rule: Integrate to obtain $\frac{5}{9} t^{9/5} + C$
Back-substitute: Replace $t$ with $(x^{2} + 7 x + 3)$ to express the answer in terms of $x$

3.3 Q-3

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

3.3 Q-3

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps