Evaluate the following indefinite integrals. 1. \displaystyle\int ( x + 2 ) \sqrt [ 4 ] { 3 x ^ {...

Question:

Evaluate the following indefinite integrals.

1. {eq}\displaystyle\int ( x + 2 ) \sqrt [ 4 ] { 3 x ^ { 2 } + 12 x - 5 } d x {/eq}

2. {eq}\displaystyle\int p ^ { 2 } \sqrt [ 3 ] { p ^ { 3 } + 1 } d p {/eq}

Integrals:

In both the parts of the given problem on indefinite integrals, we will make use of the substitution method to simplify the given functions, before proceeding for the integration.

Answer and Explanation:


1. {eq}\displaystyle\int ( x + 2 ) \sqrt [ 4 ] { 3 x ^ { 2 } + 12 x - 5 } d x {/eq}

Multiplying and dividing the above function by 6, we get:

{eq}\begin{align*} \ & = \frac{1}{6}\int 6( x + 2 ) \sqrt [ 4 ] { 3 x ^ { 2 } + 12 x - 5 } \ d x \\ \\ \ & = \frac{1}{6}\int ( 6x + 12 ) \sqrt [ 4 ] { 3 x ^ { 2 } + 12 x - 5 } \ d x \end{align*} {/eq}

Put,

{eq}\begin{align*} \ & (3 x ^ { 2 } + 12 x - 5) = t \end{align*} {/eq}

Differentiating the above, we get:

{eq}\begin{align*} \ & (6x + 12 x) \ dx = dt \end{align*} {/eq}

Substituting the above values, we get:

{eq}\begin{align*} \ & = \frac{1}{6}\int (t)^{1/4} \ dt \\ \\ \ & = \frac{1}{6} \cdot \frac{(t)^{5/4}}{\frac{5}{4}} + c\\ \\ \ & = \frac{1}{6} \cdot \frac{4}{5} \cdot (t)^{5/4}+ c \\ \\ \ & = \frac{2 t^{5/4}}{15} + c \end{align*} {/eq}

Substituting the value of {eq}t {/eq} back, we get:

{eq}\begin{align*} \ & = \frac{2 (3 x ^ { 2 } + 12 x - 5)^{5/4}}{15} + c \end{align*} {/eq}


2. {eq}\displaystyle \int p ^ { 2 } \sqrt [ 3 ] { p ^ { 3 } + 1 } \ dp {/eq}

Put,

{eq}\begin{align*} \ & (p ^{3} + 1) = t \end{align*} {/eq}

Differentiating the above, we get:

{eq}\begin{align*} \ & 3p^2 \ dp = dt \end{align*} {/eq}

Substituting the above values, we get:

{eq}\begin{align*} \ & = \frac{1}{3} \int (t)^{1/3} \ dt \\ \\ \ & = \frac{1}{3} \cdot \frac{ (t)^{4/3}}{\frac{4}{3}} + c \\ \\ \ & = \frac{1}{3} \cdot \frac{3}{4} \cdot (t)^{4/3} + c \\ \\ \ & = \frac{ t^{4/3}}{4} + c \end{align*} {/eq}

Substituting the value of {eq}t {/eq} back, we get:

{eq}\begin{align*} \ & = \frac{ (p ^{3} + 1)^{4/3}}{4} + c \end{align*} {/eq}


Learn more about this topic:

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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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