    Question

# ∫12+3 sinxdx

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Solution

## The correct option is D The above problem is of the form ∫1a+b sinxdx and we saw that, to integrate these kind of forms we’ll use the half angle formula of sinx in terms of tan(x2) Which is sinx=2tanx21+tan2x2 We’ll substitute tanx2=t Let’s use the same method here So, the given integral would be = ∫12+3.2tanx21+tan2x2dx =∫1+tan2(x2)2(1+tan2x2)+6tanx2dx=∫sec2(x2)2(1+tan2x2)+6tanx2dx Let’s substitute tanx2 = t We get 12sec2(x2).dx=dt So, we’ll have the integrals in terms of t - ∫22+2t2+6tdtOr∫11+t2+3tdt We can see that the integral we have now has a quadratic expression in the denominator. We have solved these kind of integrals earlier also by making the quadratic expression a perfect square. We’ll do the same here - ∫1t2+3t+94−54dtOr ∫1(t+32)2−54dt This is in the form of ∫1x2−a2dx And we can directly use the corresponding formula. Which is - ∫1x2−a2dx=12alog∣∣x−ax+a∣∣+C On using this formula we’ll get √42√5.log∣∣ ∣∣t+32−√54t+32+√54∣∣ ∣∣+C Let’s substitute the value of t in the above expression. 1√5.log∣∣ ∣∣tanx2+32−√54tanx2+32+√54∣∣ ∣∣+COr 1√5.log∣∣ ∣∣tanx2+32−√54tanx2+32+√54∣∣ ∣∣+C  Suggest Corrections  0      Similar questions  Related Videos   Integration of Trigonometric Functions
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