# Find the indefinite integral by making the substitution x = 7 tan (theta). integral {x^3} / {49 +...

## Question:

Find the indefinite integral by making the substitution {eq}x = 7 \tan (\theta) {/eq}.

{eq}\displaystyle \int \dfrac {x^3} {49 + x^2}\ dx {/eq}.

## Indefinite Integration:

• Gottfried Wilhelm Leibniz introduced this concept along with Issac Newton in the 17th century. We deal with the concepts of trigonometry and integration.
• Let {eq}\displaystyle f : I \rightarrow R {/eq}. Suppose that f has an antiderivative on F on I. Then we say f has on integral on I and for any real constant c, we call F+c an indefinite integral of f over I, denote it by {eq}\displaystyle \int f(x)dx {/eq} and read it as {eq}\displaystyle \textbf{Integral f(x) dx} {/eq}. Thus we have: {eq}\displaystyle \int f(x)dx = F(x) + c {/eq}, where c is called constant of integration.

To deduce the given problem, we use essential formulae like:

{eq}\displaystyle *\int x^a dx = \frac{x^{a+1}}{a+1} + c \quad \text{[c is the constant of integration]} \\ *\sec^2 x - \tan^2 x = 1 {/eq}

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Given:

{eq}\displaystyle \int \frac{x^3}{\sqrt{x^2 + 49}} d..(1) {/eq}

Let's assume:

{eq}\displaystyle \begin{align} x &= 7\tan(\theta)...