We have an answer to "Use any method to evaluate the integral. ∫x25+4x2dx"

Maaghu

Answered question

2021-11-17

Use any method to evaluate the integral.

\int \frac{x}{25 + 4 x^{2}} d x

Charles Clute

Beginner2021-11-18Added 17 answers

Step 1
To evaluate the given integral
Step 2
given integral is

\int \frac{x}{25 + 4 x^{2}} d x

put

u = 25 + 4 x^{2}

du = 8xdx

x d x = \frac{1}{8} d u

= \frac{1}{8} \int \frac{1}{u} d u

= \frac{1}{8} \ln | u |

= \frac{1}{8} \ln (25 + 4 x^{2}) + c

Stephanie Mann

Beginner2021-11-19Added 25 answers

Step 1: Use Integration by Substitution.
Let

u = 25 + 4 x^{2}, d u = 8 x d x, t h e n x d x = \frac{1}{8} d u

Step 2: Using u and du above, rewrite

\int \frac{x}{25 + 4 x^{2}} d x

\int \frac{1}{8 u} d u

Step 3: Use Constant Factor Rule:

\int c f (x) d x = c \int f (x) d x

\frac{1}{8} \int \frac{1}{u} d u

Step 4: The derivative of

\ln x i s \frac{1}{x}

\frac{\ln u}{8}

Step 5: Substitute

u = 25 + 4 x^{2}

back into the original integral.

\frac{\ln (25 + 4 x^{2})}{8}

Step 6: Add constant.

\frac{\ln (25 + 4 x^{2})}{8} + C

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