# 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.

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}